Professor, Denis Darpy
email: denis.darpy@dauphine.fr

Isabelle Prim-Allaz
email: isabelle.prim-allaz@univ-lyon2.fr

and relationship proneness on relationships marketing programmes

Potential effects of psychological reactance and relationship orientation on relationships marketing programmes.

Summary

This conceptual article presents the current criticisms to relationship marketing and important concepts developed in marketing and psychology to understand consumer reactance toward services packaged as formal contracts. A first part presents the numerous questions that researches are raising about the willingness of people to commit to services and presents four key variables to understand the customer's willingness to enter and develop business relationships. Indeed, it is shown that psychological reactance and relationship proneness are interesting concepts to revisit the relationship marketing paradigm and to better understand trust and commitment. A second section develops an integrative conceptual framework of the different concepts introduced in this research. We then conclude with avenues for future research.

1.Introduction

Nowadays, more and more firms are implementing relationship marketing strategies, looking to have fewer but more intense relationships with their channel partners or customers. This trend is driven by both efficiency and effectiveness concerns [START_REF] Geysken | Generalizations about trust in marketing channel relationships using meta-analysis[END_REF] but if relationship marketing is powerful in theory, it is troubled in practice [START_REF] Fournier | Preventing the premature death of relationship marketing[END_REF].

"To prevent it premature death", it is now welcome to build new strategies on the idea that not all customers are willing to enter a deep and long-term relationship (e.g., [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF]. Marketers may have forgotten that relationships take two. "It is possible that we haven't looked close enough to see that the consumer is not necessarily a willing participant in our relationship mission" (Fournier et al., 1998, p. 44). « The number of one-to-one relationships that companies ask consumers to maintain is untenable. As a result, many marketing initiatives seem trivial and useless instead of unique and valuable" (Fournier et al., 1998, p.46). The control that companies try to have on relationship through, for example, consumer data collection and cross selling is also simultaneously experienced by the customer as a loss of control [START_REF] Fournier | Preventing the premature death of relationship marketing[END_REF]. As a consequence, freedom of choice may be interpreted as a bind of commitments. Customers being more and more aware of the fact that companies are trying to reduce their choice set in order to keep them may become reactant to any new formal commitment. Besides, in order to get customer's commitment, firms try to generate dependence through investments and changing costs. Instead of that a more social line of thought would suggest the generation of social norms and trust. This could be more appropriate to organise long lasting exchange through the establishment of affective bonds between customers and their providers. As a consequence, the development of relationships based on trust and commitment should be more adequate than the development of relationships based on contractual safeguards [START_REF] Gutierrez | The consumer's relational commitment: Main dimensions and Antecedents[END_REF].

At the same time, it would be unrealistic to consider that the relational approach applies to all circumstances and to all individuals. As noted by [START_REF] Bahia | Some antecedents and consequences of the client's relational predisposition: An application to the retail banking industry[END_REF], some customers may prefer approaches that are based on the minimization of costs and interactions such as those offered by internet discount brokers. We may then suggest that all customers are not willing to engage in long term relationships [START_REF] Barnes | Closeness, strength, and satisfaction: Examining the nature of relationships between providers of financial services and their retail customers[END_REF] and that firms should respect it in order to develop relevant and valuable exchanges. The relationship marketing literature does not specify clearly what is the role that individual variables such as relationship proneness and psychological reactance play in the process of developing trust and commitment.

The general goal of this research is to identify individual characteristics allowing to better explain the levels of trust and commitment that customers give to their service providers.

Especially, we try to investigate the relationship between trust and commitment, considering individual variables such as relationship proneness and psychological reactance.

Literature review

Even though it is now well accepted that relationship marketing is not a new paradigm but a necessary complementary approach to "traditional marketing" (e.g., [START_REF] Kumar | Antecedents and Consequences of Relationship Intention: Implications for Transaction and Relationship Marketing[END_REF], the research literature remains very discrete on defining cases in which one or the other approach is more relevant (e.g., [START_REF] Reinartz | On the Profitability of Long-life Customers in a Noncontractual Setting: An Empirical Investigation and Implication for Marketing[END_REF].

Very little empirical work aims at explaining how some individual characteristics could be related to trust and commitment [START_REF] Mathieu | A review and meta-analysis of the antecedents, correlates and consequences of organizational commitment[END_REF]. As [START_REF] Ganesan | Negotiation strategies and the nature of channel relationships[END_REF] distinguishes between the duration of the relationship and the intention to pursue it, we propose to distinguish the intention of entering a new relationship from the intention to maintain it. To do so, there are in the relationship marketing and psychological literatures two interesting variables which could be helpful at explaining why customers do not want to enter a relationship (psychological reactance) and why they desire or not to maintain a relationship (relationship proneness).

Psychological reactance refers to the intention to enter or not a formal relationship. Another approach to relationship is to take into consideration the relationship life cycle. For example, [START_REF] Dwyer | Developing Buyer-Seller Relations[END_REF] consider that there are at least 5 stages in the relationship process:

Figure 1

Relationship proneness which refers to the intention to continue or not a relationship (formally or not) may be then useful to explain the transition from exploration to commitment.

The simultaneous use of relationship proneness and psychological reactance may seem paradoxical. We believe it is not if we consider the two constructs in a temporal perspective as defined above. We also believe that some customers may be reactant (they do not want to

Awareness Exploration Expansion Commitment Dissolution

From [START_REF] Dwyer | Developing Buyer-Seller Relations[END_REF] enter a formal relationship) and at the same time relational prone (they remain very loyal to their providers in a informal relationship).

Relationship marketing is built on two main constructs, trust and commitment. Indeed, it recommends the generation of trust and the establishment of mutual commitment as a relevant way to build successful and long-lasting relationships [START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF]. There must be a clear desire of continuity and efficient relationships should not be only based only on contractual and legal mechanisms [START_REF] Nevin | Relationship marketing and distribution channels: exploring fundamental issues[END_REF]. As mentioned by [START_REF] Gutierrez | The consumer's relational commitment: Main dimensions and Antecedents[END_REF], trust and commitment have mainly been applied to industrial markets but they can also be useful to better understand relationships in consumer-service provider relationships.

Psychological reactance

As mentioned in the introduction, relationship marketing is widely based on commitment [START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF]. This commitment can be either contractual or not. In the first case, the formalization of the commitment may be seen by customers as a formal and indefeasible tie which can be interpreted as a threat to their freedom of choice as for some promotional influence or manipulative advertisement [START_REF] Clee | Consumer behaviour and psychological reactance[END_REF]. The contractualization may be seen by some customers as a threat to their freedom for future choices. As a consequence, some customers develop strategies to reply to this feeling of freedom reduction through a commitment refusal. Actually, customers may have different cognitive, affective and behavioural responses for different levels of perceived decision freedom.

Psychological reactance is "the motivational state that is hypothesized to occur when a freedom is eliminated or threatened with elimination" (Brehm and Brehm, 1981, p. 37). The theory indicates that when a perceived freedom is eliminated or threatened with elimination, the individual will be motivated to re-establish that freedom. When an individual perceives a specific freedom, any force on the individual that makes it more difficult for him or her to exercise that freedom constitutes a threat [START_REF] Brehm | A theory of Psychological Reactance[END_REF][START_REF] Brehm | Psychological Reactance[END_REF].

Psychological reactance is associated with defensiveness, dominance and aggressiveness [START_REF] Dowd | Motivational components of client reactance[END_REF]. Reactant people have a tendency to act without considering potential consequences [START_REF] Buboltz | Personality and psychological reactance: extending the nomological net[END_REF]. For reactance to occur, the individual must perceive the freedom in question as being important [START_REF] Clee | Consumer behaviour and psychological reactance[END_REF][START_REF] Lessne | Reactance theory in consumer research : the past, present and future[END_REF].

In persuasion models, psychological reactance is presented as a mediator between communication and attitude/behaviour [START_REF] Dillard | On the Nature of Reactance and its Role in Persuasive Health Communication[END_REF]. Direct restoration of freedom involves doing the forbidden act. In addition, freedom may be restored indirectly by increasing liking for threatened choice, derogating the source of threat, denying the existence of threat or by exercising a different freedom to gain feeling of control and choice [START_REF] Dillard | On the Nature of Reactance and its Role in Persuasive Health Communication[END_REF].

Persuasive attempts of all sorts, including public health campaigns, often fail to produce the desired effect. In some cases, they even produce results directly at odds with their intents. The theory of psychological reactance provides one theoretical perspective through which these miscarriages might be understood. The theory contends that any persuasive message may arouse a motivation to reject the advocacy. That motivation is called reactance [START_REF] Dillard | On the Nature of Reactance and its Role in Persuasive Health Communication[END_REF]. From this inception to the present, the theory may be called upon to explain resistance to long-term commitment. For reactant people, any lack of alternatives, high switching costs or long term contracts represent a threat to their freedom.

Relationship proneness

Relational behaviours such as cooperative intention generally produce strong buyer-seller bonds [START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF]. Relationships are more likely to be profitable when the clients are predisposed to receive and appreciate relational treatment [START_REF] Bahia | Some antecedents and consequences of the client's relational predisposition: An application to the retail banking industry[END_REF].

Relational oriented customers are likely to value a customized service, mutual collaboration, confidentiality and long-term perspectives [START_REF] Bahia | Some antecedents and consequences of the client's relational predisposition: An application to the retail banking industry[END_REF]. Relationship proneness is defined as a personality trait that reflects a consumer's relatively stable and conscious tendency to engage in relationships with sellers of a particular product category (De Wulf et al., 2001;[START_REF] Wulf | Investments in Consumer Relationships: A Cross-Industry Exploration[END_REF][START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF]. As relationship proneness has mostly been investigated in product-oriented contexts, other very closed concepts have been introduced in services marketing studies such as client's relational predisposition [START_REF] Bahia | Some antecedents and consequences of the client's relational predisposition: An application to the retail banking industry[END_REF]. Nevertheless, definitions remain very similar and we will retain this of [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF].

From a service encounter perspective, consumer relationship proneness can be expected to play an important role in the different service contexts, mainly when these services are built on people-based service encounters. As suggested by [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF] with hairdressers, front-line employees are an important source of social support and assistance to clients with personal problems.

Given the need of both parties to maintain a relationship, there is a support to the idea that client's relationship proneness will impact the willingness to trust the partner and to commit in long term relationships. [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF] study the relationship between relationship proneness and commitment and demonstrates the existence of a positive correlation in the hairdresser context 1 .

To our knowledge, there is no study considering simultaneously relationship proneness and psychological reactance.

Trust

Trust is a major determinant of commitment (e.g., [START_REF] Achrol | Evolution of the marketing organization: New forms for turbulent environements[END_REF]Moorman et al., 1992;[START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF] and is conceptualized as follows: trust exists "when one party has confidence in an exchange partner reliability and integrity" (Morgan and Hunt, 1994, p. 23). [START_REF] Geysken | Generalizations about trust in marketing channel relationships using meta-analysis[END_REF] find that trust contributes to satisfaction and long-term orientation over and beyond the effects of economic outcomes of the relationship.

Trust is often defined as being two-dimensional, even though major references still use unidimensional measures (e.g., [START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF]. Indeed, more developed definitions insist on the fact that trust means that one believes that its exchange partner is honest and/or benevolent [START_REF] Geysken | Generalizations about trust in marketing channel relationships using meta-analysis[END_REF]. Some fellows add a third dimension: competence [START_REF] Mayer | An integrative model of organizational trust[END_REF][START_REF] Christopher | Relationship marketing: Bringing quality, customer service and marketing together[END_REF]. Trust in the partner honesty is the belief that one's partner is reliable, stands by its words and is sincere (Anderson and [START_REF] Narus | A model of distributor firm and manufacturer firm working partnerships[END_REF][START_REF] Dwyer | Developing Buyer-Seller Relations[END_REF]. Trust in partner's benevolence is the belief that its partner is genuinely interested 1 Bloemer et al. (2003) measure of commitment is unidimensional and closed to loyalty. in one's interests or welfare. A benevolent partner balances immediate self-interest with long range group gain [START_REF] Crosby | Relationship quality in services selling : An interpersonal influence perspective[END_REF]. Trust in the partner competence or credibility is based on the extent to which one believes that its partner has the required expertise to perform the job effectively and reliably [START_REF] Ganesan | Negotiation strategies and the nature of channel relationships[END_REF].

Trust is now well accepted as playing a central role in relationship building and maintenance (e.g., [START_REF] Dwyer | Developing Buyer-Seller Relations[END_REF][START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF][START_REF] Geysken | Generalizations about trust in marketing channel relationships using meta-analysis[END_REF]. Trust is supposed to lead to cooperative behaviors and commitment [START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF][START_REF] Geysken | Generalizations about trust in marketing channel relationships using meta-analysis[END_REF]. [START_REF] Geysken | Generalizations about trust in marketing channel relationships using meta-analysis[END_REF] find that trust is a key mediator variable, influencing satisfaction and long term orientation over and beyond the economic outcomes of the relationship.

Commitment

Cook andEmerson (1978, p. 728 in Morgan andHunt, 1994, p.23) characterize commitment as "a variable [they] believe to be central in distinguishing social from economic exchanges".

Commitment can be defined as "an enduring desire to maintain a valued relationship" (Moorman et al., 1992, p. 316). This means that "the committed party believes the relationship is worth working on to ensure that it endures indefinitely" (Morgan and Hunt, 1994, p. 23). Moreover, [START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF] consider that relationship commitment can only exist when the relationship is considered important. Moreover, [START_REF] Berry | Marketing services[END_REF] have pointed out that in the services relationship marketing area, relationships are built on the foundation of mutual commitment.

Commitment is also a key in explaining loyalty behaviours in order to make the difference between simply repeat buying -which may be the translation of a spurious loyalty-and true loyalty -which is closely linked with a positive attitude toward the service provider- [START_REF] Day | A two-dimensional concept of brand loyalty[END_REF].

Commitment in Business-to-consumer contexts often creates dependence asymmetry. While there is a dependence asymmetry in the relationship or a suspected one in a potential relationship, customers may be suspicious. In these cases, literacy has found that dependence may lead to calculative commitment and that the relationship will probably be dissolved as soon as the obligation derived from the dependence will end up (Iacobucci and Ostrom, 1996;Geysken et al., 1996).

Model and hypothesis:

The literature review invites us to propose a new conceptual model in order to better understand customers' positive and/or negative reactions to relationship marketing programmes.

As said previously, the theory of psychological reactance suggests that reactant people are always trying to keep or to re-establish their freedom [START_REF] Brehm | Psychological Reactance[END_REF]. This conducts us to the first hypothesis:

H1: The higher the psychological reactance, the lower the commitment.

As a consequence, reactant people should less enter into formal or contractual relationship than non reactant people.

As demonstrated by [START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF], relational behaviours such as cooperative intention generally lead to strong bonds between relationship partners. Nevertheless, all individuals may not have the same willingness to commit as explained by [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF] through their concept of relationship proneness.

As a consequence, we can formulate a second hypothesis:

H2: The higher the relationship proneness, the higher the commitment

Marketing and psychological literatures seem to never consider relationship proneness and psychological reactance at the same time. The present authors have conducted a qualitative analyses based on 20 in-depth interviews in the health sector. Results seem to show the independence between the two constructs. Indeed, we have identified different categories of respondents among which:

• High reactant and low relationship prone respondents;

• Low reactant and high relationship prone respondents;

• High reactant and high relationship prone respondents.

If the two first categories are not surprising, the third one seem to be very interesting and show a relative independence between the two constructs 2 .

The literature review has presented trust as a major determinant of relationship commitment (e.g., [START_REF] Achrol | Evolution of the marketing organization: New forms for turbulent environements[END_REF]Moorman et al., 1992;[START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF]. It is now widely recognized as influencing relationship building and maintenance (e.g., [START_REF] Dwyer | Developing Buyer-Seller Relations[END_REF][START_REF] Morgan | The commitment-trust theory of relationship marketing[END_REF][START_REF] Geysken | Generalizations about trust in marketing channel relationships using meta-analysis[END_REF]. As a consequence, we formulate the following hypothesis:

H3: The higher the trust, the higher the commitment H4: Trust mediates the relationship between relationship proneness and commitment H4: Trust mediates the relationship between psychological reactance and commitmentAll relationships are illustrated in Figure 2. 2 We should logically consider a fourth category « low psychological reactant and low relationship prone customers". In fact this category is not really interesting as it is made of customer with a very low level of implication.

Conclusion

This conceptual research is a first step of a wider research program. Its objectives was to present current criticisms to relationship marketing and to show how important concepts developed either in the marketing or the psychological fields may of help to better understand relationship marketing failures. Psychological reactance helps us to understand the non willingness of certain customers to enter long term and contractual relationships;

relationship proneness helps us to better understand the willingness of customers to maintain or not a relationship. This research is obviously a first step, the next one being an empirical validation of the model.

Nevertheless, we can already point out theoretical and managerial implications.

On a theoretical point of view, this research proposes a new approach to understand relationship commitment through two variables: relationship proneness and psychological reactance. As long as we know, these two variables never appear simultaneously in this context. Moreover, they appear as being really complementary from each other as they concern different stages of the relationship development.

On a managerial point of view, this research should help managers to better understand their customers' behaviours and expectations in term of relationship. Developing relevant measurement tools for these two variables should allow managers to segment their customers according to their willingness to enter and to develop the relationship. Indeed, psychological reactant customers should refuse formal and long-term commitment. Relationship prone customers should be willing to commit and to develop the relationship on a long term basis.

As early mentioned one category of customers may be very interesting for managers for developing new types of relationships: the highly psychological reactant and highly relationship prone. Indeed, for the low psychological reactant and highly relationship prone customers, traditional relationship marketing tools should be efficient; for the highly psychological reactant and low relationship prone customers transactional programs offer the best solutions (Cf Annex .1, example 1). But for the third category (highly psychological reactant and highly relationship prone), neither traditional relationship marketing nor transactional tools may be efficient. This category of people seems to be quite loyal to their service providers, seems to be quite trusting but refuse to sign any type of formal commitment. The challenge is, then, to build a long term relationship with them without giving them the feeling that their freedom is restricted. This may show, for example, that policies such as exit costs are not relevant for this category of customers (Cf. annex1, example 2).

As mentioned earlier, the next step is to conduct an empirical research in different fields as for example health sector and mobile phones. We believe that this two fields are interesting for different reasons:

• First they are really different fields, even though the patient-practitioner and the customer-service provider relationships may have a lot in common. In both cases the asymmetry of information is sources of frustration and ethical questions. Indeed, both long term subscription and prevention illness programmes require commitment and disclosure of information of the customer or patient towards the practitioner or the service provider.

• They are different in term of level of implication. In the health context, the patient health is at stake which automatically increases the level of patient's implication.

• The competition is different in the two fields. The health sector is still strictly controlled in many countries, while the mobile phone sector is more competitive.

  Figure 2

This research has received financial support from the FAQSV (Fonds d'aide à la qualité des Soins en Ville) and the URCAM (Union régionale des caisses d'assurance maladie) of Ile de France. The authors also thank Doctors Thierry Slama and Jean-Luc Sabban from the GIPS network.

Appendix 1

Example 1


Roland Bacher Twisting The Stern

Roland Bacher

Twisting the Stern sequence

Keywords: Stern sequence, automatic sequence, regular sequence. Math. class: 11B85

teaching and research institutions in France or abroad, or from public or private research centers.

1 of the form 1(01) * . We have s(11) = s(5) + s(6) = s(2) + s(3) + s(3) = s(1) + 2(s(1) + s(2)) = 5s(1) = 5.

Proposition 1.1 allows to parametrize Stern sequences by counting a subsequence of the form 1(01) k with weight w k , see Proposition 2.1 for formulae. For n = 11 we get for instance 3 + 2w.

In this paper we introduce a related sequence which will be called the twisted Stern sequence since it is obtained by twisting the recursive definition of the Stern sequence with a sign. More precisely, we define the twisted Stern sequence t(0), t(1), . . . recursively by t(0) = 0, t(1) = 1 and t(2n) = -t(n), t(2n + 1) = -t(n) -t(n + 1) for n ≥ 1. It starts as 0, 1, -1, 0, 1, 1, 0, -1, -1, -2, -1, -1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, . . . . An inspection of these first few terms shows already some striking similarities between the Stern sequence and its twisted relative. The aim of this paper is to describe a few properties of the Stern sequence and its twist.

The following result (the identity for s(n) in assertion (i) is probably wellknown to the experts) is an illustration of the similarities between these two sequences:

Theorem 1.2 (i) We have s(2 e + n) = s(2 e -n) + s(n) t(2 e + n) = (-1) e (s(2 e -n) -s(n))
for all e ≥ 0 and for all n such that 0 ≤ n ≤ 2 e .

(ii) We have

t(3 • 2 e + n) = t(6 • 2 e -n) = (-1) e s(n)
for all e ≥ 0 and for all n such that 0 ≤ n ≤ 2 e+1 .

The failure for n > 2 e+1 of the formula

t(3 • 2 e + n) = (-1) e s(n)
given by assertion (ii) can perhaps be mended by the following conjectural identity based on experimental observations. Conjecture 1.3 There exists an integral sequence u(0), u(1), u(2), . . . such that we have

∞ n=0 t(3 • 2 e + n)z n = (-1) e ∞ n=0 u(n)z n•2 e ∞ m=0 s(m)z m
for all e ∈ N.

If the conjecture holds, the ordinary generating function of the sequence u(0), u(1), . . . is given by

∞ n=0 u(n)z n = ∞ n=0 t(3 + n)z n ∞ n=0 s(n)z n
and it starts as 1 -2z 2 -2z 5 + 4z 6 + 2z 7 -6z 8 + 4z 9 + 2z 10 -6z 11 + 8z 12 + . . . The first equality in assertion (ii) of Theorem 1.2 shows that the finite sequences (-1) e t(3 • 2 e ), (-1) e t(3 • 2 e + 1, . . . , (-1) e t(6 • 2 e ) of length 3 e + 1 are palindromic sequences of natural integers. The first few such sequences are

0 1 1 0 1 1 2 1 1 0 1 1 2 1 3 2 3 1 2 1 1 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 3 2 3 1 2 1 1
with boldfaced 1's at one third and two thirds highlighting the underlying partial self-similarity structure. The subsequence lying between the two boldfaced 1 ′ s appears also at the beginning of [START_REF] Lehmer | On Stern's Diatomic Series[END_REF]. One notices that all sequences start and end with zero and that all existing central elements are equal to 2.

The polynomials defined by these palindromic sequences are described by the following result:

ψ e = z(1 + z 2 e )(1 + z + z 2 ) e e-2 n=0 (1 -z 2 n + z 2 n+1 ) e-1-n (1) = z(1 + z 2 e ) e-1 n=0 (1 + z 2 n + z 2 n+1 ) . (2)
Remark 1.5 Theorem 1.4 implies the identity

∞ n=0 t(n)z n = z -z 2 + ∞ e=0 (-1) e z 3•2 e +1 (1 + z 2 e ) e-1 n=0 (1 + z 2 n + z 2 n+1 ) .
Assertion (ii) of Theorem 1.2 yields lim e→∞ ψ e = ∞ n=0 s(n)t n . The factorisation (2) of Theorem 1.4 gives a new proof of the following result due to Carlitz (see [START_REF] Carlitz | A problem in partitions related to the Stirling numbers[END_REF]):

Corollary 1.6 We have ∞ n=0 s(n)z n = z ∞ n=0 (1 + z 2 n + z 2 n+1 ).
(A direct proof of Corollary 1.6 is straightforward: The series U (z) defined by the right-hand-side starts as z +. . . = s(0)+s(1)z +. . . and its even, respectively odd, subseries are given by U (z 2 ), respectively 1 z + z U (z 2 ). Its coefficients satisfy thus the same recursion relations as the elements of the Stern sequence.)

The Carlitz factorisation of Corollary 1.6 implies that ∞ n=0 s(n)z n has no non-zero roots in the open unit disc. This is not true for the ordinary generating series ∞ n=0 t(n)z n of the twisted Stern sequence which has (infinitely?) many non-zero roots in the open unit disc.

Given a natural integer k ≥ 2 and a natural integer i, we consider the endomorphism ρ(i) of the vector-space (or module) of formal power series defined by

ρ(i) ∞ n=0 a(n)z n = ∞ n=0 a(i + nk)z n .
The k-kernel of a formal power series A is the smallest vector space (or module when working over a ring) V containing A such that ρ(0)V, . . . , ρ(k -1)V ⊂ V. A formal power series is k-regular if its k-kernel is finitely generated. Easy examples of k-regular series are polynomials and ordinary generating series of periodic sequences. k-regular power series form a vector space (or module) which is preserved by many natural operations such as derivation, product, Hadamard product, shuffle product, ..., see [START_REF] Allouche | Automatic Sequences. Theory, Applications, Generalizations[END_REF] for details. The set of k-regular sequences with coefficients contained in a finite set (eg. in a finite field) coincides with the set of so-called k-automatic sequences, see Theorem 16.1.5 of [START_REF] Allouche | Automatic Sequences. Theory, Applications, Generalizations[END_REF].

A sequence a(0), a(1), . . . is called k-regular if its ordinary generating series ∞ n=0 a(n)z n is a k-regular formal power series. The following result is also a consequence of the Carlitz factorisation: Theorem 1.7 The logarithmic derivative

H(z) = d dz log ∞ n=0 s(n + 1)z n = ∞ n=1 ns(n + 1)z n-1 ∞ n=0 s(n + 1)z n of ∞ n=0 s(n + 1)z n is 2-regular.
More precisely, H(z) is defined by the functional equation

H(z) = 1 + 2z 1 + z + z 2 + 2zH(z 2 ) .
Coefficients of H(z) appear as sequence A163659 in [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF]. 2-regularity of the logarithmic derivation H(z) is a special case of the following result, perhaps already known to Schützenberger: Theorem 1.8 Given d k-regular series A 1 (z), . . . , A d (z) over some commutative ring R, d linear forms L 1 (x 1 , . . . , x d ), . . . , L d (x 1 , . . . , x d ) in d unknowns with coefficients in R[z] and d constants α 1 , . . . , α d ∈ R such that

α i = A i (0) + L i (α 1 , . . . , α d ) (mod z) for i = 1, . . . , d, the system of equa- tions U 1 (z) = A 1 (z) + L 1 (U 1 (z k ), . . . , U d (z k )), . . . U d (z) = A n (z) + L n (U 1 (z k ), . . . , U d (z k ))
determines a unique set of d k-regular sequences U 1 (z), . . . , U d (z) with constant coefficients

α i = U i (0) for i = 1, . . . , d. Remark 1.9 We have α i = A i (0) if the linear form L i has all its coeffi- cients in zR[z].
The series

A(z) = ∞ n=0 1 1 -z 2 n satisfying A(z) = ( ∞ n=0 z n ) A(z 2
) is not 2-regular (see Remark 1.11 below). This shows that Theorem 1.8 can not be extended to equations with linear forms having k-regular series as coefficients.

Theorem 1.7 can be generalised as follows:

Theorem 1.10 Let P (z) be a polynomial with constant coefficient 1. Then the series

A = ∞ n=0 P (z k n ) is k-regular.
Moreover, if all roots of P (z) are complex roots of 1 having finite order, then the logarithmic derivative

B = A ′ /A of A is also k-regular. Remark 1.11 Given a k-regular series A(z) with constant coefficient 1, the product ∞ n=0 A(z k n ) is generally not k-regular. Indeed, starting with the 2-regular series A(z) = 1 + z + z 2 + . . . = 1 1-z , the coefficient b(n) in the series B = ∞ n=0 b(n)z n = ∞ n=0 1 1 -z 2 n
counts the number of partitions of n into powers of 2, see sequence A123 in [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF], and log(b(2n)) is asymptotically equal to 1 2 log 2 log n log n

2

(see equation 1.3 in [START_REF] De Bruijn | On Mahler's partition problem[END_REF]) which is incompatible with k-regularity of B by Theorem 16.3.1 in [START_REF] Allouche | Automatic Sequences. Theory, Applications, Generalizations[END_REF].

Remark 1.12 Another famous sequence illustrating Theorem 1.10 is the sequence

∞ n=0 (1 -z 2 n ) = ∞ n=0 (-1) tm(n) z n
related to the Thue-Morse sequence n -→ tm(n) defined by digit-sums modulo 2 for binary expansions of natural integers.

Another link between the two sequences s and t is given by determinants of 2 × 2-matrices. For n ≥ 1 we consider the matrix

M (n) = s(n) s(n + 1) t(n) t(n + 1)
with first row two consecutive terms of s and and second row the two corresponding consecutive terms of t. The first matrices are 2 × 2-submatrices defined by two consecutive rows of

1 1 2 1 3 2 3 1 4 3 5 2 5 3 5 1 5 4 7 1 -1 0 1 1 0 -1 -1 -2 -1 -1 0 1 1 2 1 3 2 3
Injectivity of the map n -→ s(n + 1)/s(n) shows that the matrices M (n) are all different. Their determinants are characterised by the following result.

Theorem 1.13 We have | det(M (n))| = 2 for all n ≥ 1. More precisely, det(M (n)) = -2(-1) k if 2 k ≤ n < 2 k+1 .
In particular, all matrices M (n) are invertible for n ≥ 1.

Since the recursive definitions of the integral sequences s and t differ only by signs (and since they satisfy the same initial conditions) they have the same reduction modulo 2 characterised by the following (easy) result, already contained in [START_REF] Stern | Über eine zahlentheoretische Funktion[END_REF]: Proposition 1.14 The integers s(n) and t(n) are even if and only if n is divisible by 3.

The determinants of the non-singular matrices M (n) are thus in some sense as small as possible: Indeed, since the sequences s and t coincide modulo 2, a matrix M (n) involves either a column consisting of even integers or all its four entries are odd integers and such matrices have even determinants.

We end this paper with a last result, going back to Stern (see [START_REF] Stern | Über eine zahlentheoretische Funktion[END_REF]) for the first part of assertion (i): Theorem 1.15 (i) The integer s(n) divides s(n -1) + s(n + 1) for every n ≥ 1. More precisely, we have

s(n -1) + s(n + 1) s(n) = 1 + 2v 2 (n)
where the 2-valuation v 2 (n) is defined as the exponent of the highest power of 2 dividing n.

The function

C(z) = ∞ n=1 s(n -1) + s(n + 1) s(n) z n
satisfies C(0) = 0 and

C(z) = z 1 + 2z 1 -z 2 + C(z 2 )
and is 2-regular.

(ii) The integers t(n) and t(n -1) + t(n + 1) are both zero if n = 3 • 2 k . They are both non-zero otherwise and t(n) divides t(n -1) + t(n + 1). More precisely, we have

t(n -1) + t(n + 1) t(n) = 1 + 2v 2 (n) if n ∈ {2 N , 3 • 2 N }, (t(0) + t(2))/t(1) = -1 and t(2 e -1) + t(2 e + 1) t(2 e ) = 1 + 2(e -2)
for all e such that e ≥ 1.

The rest of this paper is organized as follows: The next section contains a proof of Proposition 1.1 and a few complements.

Section 3 is devoted to the (easy) proofs of Theorem 1.2 and to a few more formulae and conjectures involving the Stern sequence and its twist.

Section 4 contains the proofs of Theorems 1.4, 1.7, 1.8, 1.10 and Corollary 1.6.

Section 5 contains the easy proof of Theorem 1.13 and a few related results.

Section 6 consists of the short proof of Theorem 1.15.

Proof of Proposition 1.1 and a few comments

We give first a bijective proof of Proposition 1.1. We describe then briefly a weighted version of the Stern sequence counting subsequences of the form 1(01) * with weights encoding their length. We give also a generalisation of Proposition 1.1 therefore providing a (sketch of a) second proof for Proposition 1.1.

Proof of Proposition 1.1 We call a subsequence of a binary expansion B(n) admissible if it is of the form 1(01) * . For example, B(11) = 1011 = ǫ 3 ǫ 2 ǫ 1 ǫ 0 has five admissible subsequences given by the set

{ǫ 3 , ǫ 1 , ǫ 0 , ǫ 3 ǫ 2 ǫ 1 , ǫ 3 ǫ 2 ǫ 0 } .
Since the number b(n) of such subsequences satisfies clearly b(2n) = b(n), the equality s(2n) = s(n) shows that we can restrict our attention to n odd. We consider the two cases n = 4n + 1 and n = 4n -1.

If w is an admissible subsequence of B(4n + 1), then the digit ǫ 1 = 0 of the binary expansion of 4n + 1 is either contained in w or not. In the first case, admissibility of w shows that w contains also the last digit ǫ 0 of B(4n + 1) and removal of ǫ 1 ǫ 0 = 01 from w yields a bijection between such admissible subsequences and admissible subsequences of B(2n). In the second case where ǫ 1 is not involved in w, we get a bijection between such admissible subsequences and admissible subsequences of the binary expansion B(2n + 1) = . . . ǫ 3 ǫ 2 ǫ 0 of the integer 2n + 1. This shows the identity b(4n + 1) = b(2n) + b(2n + 1).

We consider finally the case of an admissible subsequence w of B(4n-1). If ǫ 0 is not contained in w, then w can be associated with an admissible subsequence of B(4n-2) or equivalenty of B(2n-1). Denoting by l the least integer such that B(4n-1) = α01 l , we consider now an admissible subsequence w of B(4n -1) which contains ǫ 0 . If the admissible subsequence w is not of the form βǫ l ǫ 0 , we transform it into the admissible subsequence βǫ l = β1 of B(4n) = αǫ l ǫl-1 . . . ǫ0 = α10 l or equivalently of B(2n) = α10 l-1 (obtained from B(4n) by erasing the last digit 0 never involved in an admissible subsequence). If w = βǫ l ǫ 0 we transform it into the admissible subsequence β of B(4n) or equivalently of B(2n). This shows b(4n -1) = b(2n -1) + b(2n) and ends the proof. The initial values for S(n) and S e (n) in assertion (ii) are easy to check. The identity S(2n) = S(n) is obvious since admissible subsequences of B(2n) never involve the last digit ǫ 0 = 0 in the binary expansion B(2n) of 2n.

Admissible subsequences of B(2n + 1) not containing the last digit ǫ 0 of B(2n + 1) are in weight-preserving bijection with admissible subsequences of B(2n) or of B(n). Removal of ǫ 0 induces a weight-preserving bijection between admissible subsequences of B(2n + 1) involving the last digit ǫ 0 of B(2n + 1) and monomial contributions to S e (n). This proves S(2n + 1) = S(n) + S e (n).

Monomial contributions to S e (2n) not involving the last digit ǫ 0 of B(2n) are in (weight-preserving) bijection with monomial contributions to S e (n). Removing the last digit of monomial contributions to S e (2n) involving the last digit ǫ 0 of B(2n) yields admissible subsequences of B(n) with weight reduced by 1. This shows S e (2n) = S e (n) + wS e (n).

The identity S e (2n + 1) = S e (n) is due to the fact that monomial contributions to S e (2n + 1) never involve the last digit ǫ 0 = 1 of B(2n + 1).

Assertion (iii) follows from the bijections used in the proof of Proposition 1.1. We leave the details to the reader. 2

Counting weighted subsequences and subfactors

A famous result by Schützenberger implies essentially an identification of k-regular sequences with the set of rational formal power series in k noncommuting variables. (One has to be a little careful with leading zeros. A way of dealing with them is to consider only formal power series involving no monomials starting with the variable x 0 associated to the digit 0.) Proposition 1.1 is then a particular case of the following well-known result which we give without proof. (A proof of stronger statements can be found in [START_REF] Berstel | Noncommutative Rational Series with Applications[END_REF].) Proposition 2.3 Let A be a rational formal power series in k non-commuting variables. Then the shuffle product of A with 1 1-(x 0 +...+x k ) and the ordinary non-commutative product

1 1-(x 0 +...+x k ) A 1 1-(x 0 +...+x k ) are both rational.
The shuffle-product counts subsequences encoded and weighted by A in k-ary expansions of natural integers and the ordinary product in Proposition 2.3 counts subfactors (encoded by A) in k-ary expansions.

Proposition 1.1 corresponds to the case where

A = x 1 1 1 -x 0 x 1 = x 1 + x 1 x 0 x 1 + x 1 x 0 x 1 x 0 x 1 + x 1 x 0 x 1 x 0 x 1 + . . . respectively A = x 1 w 1 -x 0 x 1 = x 1 + wx 1 x 0 x 1 + w 2 x 1 x 0 x 1 x 0 x 1 + . . .
in the weighted case with x 0 , x 1 non-commuting variables and w a central variable.

A famous example counting subsequences or subfactors (reduced to 1) is given by the Thue-Morse sequence corresponding to A = x 1 .

Another famous example counting subfactors is given by the Rudin-Shapiro sequence associated to A = x 2 1 .

3 Proof of Theorem 1.2 and more formulae 3.1 Proof of Theorem 1.2

Proof of assertion (i) For n = 0 we have

s(1 + 0) = s(1) + s(0) = 1 + 0 = 1, s(1 + 1) = s(0) + s(1) = 0 + 1 = 1 and t(1 + 0) = s(1) -s(0) = 1 -0 = 1, t(1 + 1) = s(0) -s(1) = 0 -1 = -1 .
The proof is now by induction on e. If n is even we have

s(2 e + n) = s(2 e-1 + n 2 ) = s(2 e-1 - n 2 ) + s( n 2 ) = s(2 e -n) + s(n) and t(2 e + n) = -t(2 e-1 + n 2 ) = -(-1) e-1 s(2 e-1 - n 2 ) -s( n 2 ) = (-1) e (s(2 e -n) -s(n))
If n is odd, we have

s(2 e + n) = s(2 e-1 + n -1 2 ) + s(2 e-1 + n + 1 2 ) = s(2 e-1 - n -1 2 ) + s( n -1 2 ) + s(2 e-1 - n + 1 2 ) + s( n + 1 2 ) = s(2 e -n) + s(n) and t(2 e + n) = -t(2 e-1 + n -1 2 ) -t(2 e-1 + n + 1 2 ) = (-1) e s(2 e-1 - n -1 2 ) -s( n -1 2 ) + s(2 e-1 - n + 1 2 ) -s( n + 1 2 ) = (-1) e (s(2 e -n) -s(n))
Proof of assertion (ii) These formulae are easy to establish for e = 0.

For even n we have

t(3 • 2 e + n) = -t(3 • 2 e-1 + n 2 ) = -(-1) e-1 s( n 2 ) = (-1) e s(n) and t(6 • 2 e -n) = -t(6 • 2 e-1 - n 2 ) = -(-1) e-1 s( n 2 ) = (-1) e s(n)
and for odd n we get

t(3 • 2 e + n) = -t(3 • 2 e-1 + n + 1 2 ) -t(3 • 2 e-1 + n -1 2 ) = -(-1) e-1 (s( n + 1 2 ) + s( n -1 2 )) = (-1) e s(n) and t(6 • 2 e -n) = -t(6 • 2 e-1 - n + 1 2 ) -t(6 • 2 e-1 - n -1 2 ) = -(-1) e-1 (s( n + 1 2 ) + s( n -1 2 )) = (-1) e s(n) .
This completes the proof. 2

A few other formulae

Proposition 3.1 (i) We have s(2 e+1 + n) = s(2 e + n) + s(n)
for 0 ≤ n ≤ 2 e , (see the remark by T. Tokita concerning the Stern-sequence A2487 in [START_REF] Sloane | The On-Line Encyclopedia of Integer Sequences[END_REF]).

(ii) We have

t(2 e+1 + n) + t(2 e + n) = (-1) e+1 s(n) for 0 ≤ n ≤ 2 e .
The formulae of Proposition 3.1 have the following conjectural generalisation, analogous to Conjecture 1.3:

Conjecture 3.2 (i) The series A(z) = ∞ n=0 (s(2 + n) -s(1 + n))z n ∞ n=0 s(n)z n = 1 -2z + 2z 2 -4z 4 + 4z 5 + 2z 6 + . . . satisfies ∞ n=0 (s(2 e+1 + n) -s(2 e + n))z n = A(z 2 e ) ∞ n=0 s(n)z n
for all e ∈ N.

Similarly, the series

B(z) = - ∞ n=0 (t(2 + n) + t(1 + n))z n ∞ n=0 s(n)z n = 1 -2z -2z 2 + 4z 3 + 6z 6 -6z 7 + . . . satisfies (-1) e+1 ∞ n=0 (t(2 e+1 + n) + t(2 e + n))z n = B(z 2 e ) ∞ n=0 s(n)z n
for all e ∈ N.

Proof of Proposition 3.1 The formulae hold for n = 0 and a ∈ {0, 1}. The induction step is an easy computation for odd a and obvious for even a.

2

Proposition 3.3 (i) We have s(n) = -s(n -2 e ) + s(n -2 • 2 e ) + 2s(n -3 • 2 e ), 2 e+2 ≤ n ≤ 2 e+3 -2 e
for e ≥ 0.

(ii) We have

t(n) = t(n -2 e ) -t(n -2 e+1 ) for 2 e+2 ≤ n ≤ 2 e+3 . Proposition 3.5 We have s(3 • 2 e + n) = s(3 • 2 e -n) (3)
for all e, n such that 0 ≤ e ≤ 2 n ,

s(3 • 2 e + n) = s(3 • 2 e-1 + n) + 2s(n) (4)
for all e, n such that 0 ≤ n ≤ 2 e-1 ,

t(2 e + n) = t(2 e + n -2 e-2 ) -t(2 e + n -2 e-1 ) (5)
for all e, n such that e ≥ 2 and 1 ≤ n ≤ 2 e , s(2 e + n) = (-1) e t(2 e + n) + 2s(n) [START_REF] Carlitz | A problem in partitions related to the Stirling numbers[END_REF] for all e, n such that 0 ≤ n ≤ 2 e+1 , s(2 e + n) = (-1) e t(2 e -n) -3s(n) [START_REF] Klavzar | Stern polynomials[END_REF] for all e, n such that 0 ≤ n ≤ 2 e-1 ,

s(2 e -n) = (-1) e t(2 e -n) + 2s(n) (8)
for all e, n such that 0 ≤ n ≤ 2 e-1 , s(2 e -n) = (-1) e t(2 e + n) + s(n)

for all e, n such that 0 ≤ n ≤ 2 e .

Proofs are easy and left to the reader.

Proofs related to factorisations

Proof of Theorem 1. [START_REF] De Bruijn | On Mahler's partition problem[END_REF] We set ψ e = z(1 + z 2 e )(1 + z + z 2 ) e-2 n=0 (1 -z 2 n + z 2 n+1 ) e-1-n . Iterating the trivial identity

(1 + z n + z 2n )(1 -z n + z 2n ) = (1 + z 2n + z 4n )
we get the equivalent expression

ψ e = z(1 + z 2 e ) e-1 n=0 (1 + z 2 n + z 2 n+1 ) .
The proof of the identity ψ e = (-1) e 3•2 e n=0 t(3 • 2 e + n)z n is by induction on e. It holds for e = 0. The induction step follows from the recursive definition of the sequence t(0), t(1), . . . and from the equality ψ e+1 (z) =

1 z + 1 + z ψ e (z 2 ). 2
we construct an equivalent system with strictly smaller maximal degree for the linear forms L 1 , . . . , L 2d . Iteration of this construction leads eventually to a system containing only linear forms of degree strictly smaller than k.

Existence and unicity of the solution follow from unicity of the attracting fixpoint of the dynamical system defined by the map

U i (z) -→ A i (z) + L i (U 1 (z k ), . . . , U d (z k )), i = 1, . . . , d
starting from the point (α 1 , . . . , α d ).

2 Proof of Theorem 1.10 The first part follows from Theorem 1.8 applied to the identity A(z) = P (z)A(z k ). We present here however a second, independent proof. Working over the field of complex numbers and using the fact that products of two k-regular series are k-regular (cf. Theorem 16.4.1 in [START_REF] Allouche | Automatic Sequences. Theory, Applications, Generalizations[END_REF]), it is enough to prove the result for polynomials of degree 1. We can thus assume that P

(z) = 1 + λz. The coefficient of z n in A(z) = ∞ m=0 (1 + λz k m
) is then given by zero if the k-ary expansion of n involves digits greater than 1 and it is given by λ α otherwise, where α equals the number of ones in the k-ary expansion of n. This implies k-regularity of A(z).

We have

B(z) = d dz log(A(z)) = n=0 P ′ (z k n )k n z k n -1 P (z k n ) .
The summand of index n = 0 yields P ′ (z) P (z) and the remaining summation ∞ n=1 • • • can be rewritten as kz k-1 B(z k ). This shows that B(z) satisfies the functional equation

B(z) = P ′ (z) P (z) + kz k-1 B(z k ) .
Using Theorem 16.4.3 of [START_REF] Allouche | Automatic Sequences. Theory, Applications, Generalizations[END_REF] we see that the rational fraction P ′ (z) P (z) is k-regular if and only if all zeroes of P (z) are roots of unity (ie. if P (z) divides (z N -1) N for some integer N ). Theorem 1.8 implies then k-regularity of B(z).

= det s(n) s(n) + s(n + 1) -t(n) -t(n) -t(n + 1) = -det s(n) s(n + 1) t(n) t(n + 1) = -det(M (n)) and det(M (2n -1)) = det s(2n -1) s(2n) t(2n -1) t(2n) = det s(n -1) + s(n) s(n) -t(n -1) -t(n) -t(n) = -det s(n -1) s(n) t(n -1) t(n) = -det(M (n -1))
imply the result. 2 Proof of Proposition 1.14 The reduction modulo 2 of the Stern sequence s(0), s(1), . . . is the 3-periodic sequence 0, 1, 1, 0, 1, 1, . . .. Indeed, this holds for s 0 = 0, s 1 = s 2 = 1 and the recursive formulae s(6n) = s(3n) s(6n + 1) = s(3n) + s(3n + 1) s(6n + 2) = s(3n + 1) s(6n + 3) = s(3n + 1) + s(3n + 2) s(6n + 4) = s(3n + 2) s(6n + 5) = s(3n + 2) + s(3n + 3) imply the 3-periodicity of s(n) (mod 2) by induction. The reduction modulo 2 of twisted Stern sequence t(0), t(1), . . . coincides with the reduction modulo 2 of the Stern sequence. 2

Other results involving matrices

The proofs of the following results are easy and omitted. (ii) The matrices s(n) s(n + 1) t(2 e + n) t(2 e + n + 1) have determinant (-1) e+1 for n such that 0 ≤ n < 2 e and determinant (-1) e for n such that 2 e ≤ n < 2 e+2 .

(iii) The matrices t(n) t(n + 1) s(2 e + n) s(2 e + n + 1) have determinant (-1) e+1 for n such that 2 e+1 < n < 5 • 2 e .

(iv) The matrices t(n) t(n + 1) t(2 e + n) t(2 e + n + 1) have determinant 1 for n such that 2 e-2 ≤ n < 2 e or 7 • 2 e ≤ n < 2 e+3 and determinant -1 for n such that 2 e ≤ n < 7 • 2 e .

6 Proof for Theorem 1.15

For odd n we have s(n) = s((n -1)/2) + s((n + 1)/2) = s(n -1) + s(n + 1) .

We have thus s(n-1)+s(n+1) s(n) This ends the proof of assertion (i).

= 1 = 1 + 2v 2 (n) since v 2 (n) = 0 if n is odd.
For the twisted Stern sequence we use the analogous identities t(n -1) + t(n + 1) = t(n), n odd, t(n -1) + t(n + 1) = -(t(n/2 -1) + t(n/2 + 1) + 2t(n)), n even, n ≥ 4.

This implies assertion (ii) by checking the initial cases and the case of n ∈ 3 • 2 N . 2

Theorem 1 . 4

 14 The polynomials ψ e = (-1) e 3•2 e n=0 t(3 • 2 e + n)z n have the factorisations

2

 2

2. 1 A

 1 weighted variation of the Stern sequenceWe denote by S(n) ∈ N[w] the weighted number of subsequences of the form 1(01) * in the binary expansion B(n) of n, giving the weigth w k to a subsequence of the form 1(01) k . Similarly, we introduce S e (n) ∈ N[w] as the weighted number of subsequences of the form (10) * in the binary expansion B(n) of n, with weigth w k for a subsequence of the form (10) k . Proposition 2.1 (i) Evaluating the polynomial S(n) ∈ N[w] at w = 1 yields the Stern sequence. (ii) The sequences S(n) and S e (n) are uniquely determined by the initial conditions S(0) = 0, S(1) = S e (0) = S e (1) = 1 and the recursive formulae S(2n) = S(n) S(2n + 1) = S(n) + S e (n) S e (2n) = wS(n) + S e (n) S e (2n + 1) = S e (n) . (iii) The sequence S(n) is also uniquely determined by the initial conditions S(0) = 0, S(1) = 1 and by the recursive formulae S(2n) = S(n) S(4n + 1) = wS(2n) + S(2n + 1) S(4n -1) = S(2n -1) + S(2m + 1) + (w -1)S(2m) where n = 2 a (2m + 1). Remark 2.2 Klavzar, Milutinovic and Petri have studied a different family of polynomials closely related to the Stern sequence by considering B 0 = 0, B 1 = 1, B 2n = tB(n) and B 2n+1 = B n + B n+1 , see [7] for details. Proof of Proposition 2.1 Assertion (i) is obvious. In the sequel, we use the notation introduced above during the proof of Proposition 1.1.

2 5

 2 Proof of Theorem 1.13 and other results involving matrices Proof of Theorem 1.13 The trivial identities det(M (2n)) = det s(2n) s(2n + 1) t(2n) t(2n + 1)

Proposition 5 . 1

 51 (i) The matrices s(n) s(n + 1) s(2 e + n) s(2 e + n + 1)have determinant -1 for n such that 0 ≤ n < 2 e and determinant 1 for n such that 2 e ≤ n < 2 e+1 .

2 = 1 +

 21 For n even we have by inductions(n -1) + s(n + 1) s(n) = s((n -2)/2) + s(n/2) + s(n/2) + s((n + 2)/2) s(n/2) = s(n/2 -1) + s(n/2 + 1) s(n/2) + 2v 2 (n/2) + 2 = 1 + 2v 2 (n) .

Proof

The case e = 0 implies n ∈ {4, 5, 6, 7} in assertion (i) and we have s(4) = 1 = -2 + 1 + 2 • 1 = -s(3) + s(2) + 2s(1) s(5) = 3 = -1 + 2 + 2 • 1 = -s(4) + s(3) + 2s(2) s(6) = 2 = -3 + 1 + 2 • 2 = -s(5) + s(4) + 2s(3) s(7) = 3 = -2 + 3 + 2 • 1 = -s(6) + s(5) + 2s [START_REF] De Bruijn | On Mahler's partition problem[END_REF] The induction step for e > 0 is easy if n is even and involves the usual identity s

The proof of assertion (ii) is similar. 2 The following result gives a few partial sums associated to the Stern sequence and its twist:

Proof The first equality holds for e = 0 and

For the next identity one finds similarly

The computations for the partial sums involving t(n) and (-1) n t(n) are analogous.

2 We end this section with a list of a few more identities.

Proof of Theorem 1.7: Using the Carlitz factorisation

The summand of index n = 0 yields 1+2z 1+z+z 2 and the sum ∞ n=1 • • • can be rewritten as 2zH(z 2 ).

The proof of 2-regularity of H(z) is an easy consequence of the functional equation for H, see Theorem 1.8 below.

Uniqueness of H defined by the functional equation H(z) = 1+2z 1+z+z 2 + 2zH(z 2 ) follows from the fact that the map

has a unique attracting fixpoint for formal power series (with respect to the obvious topology given by coefficent-wise convergency).

2 Proof of Theorem 1.8 We assume first that no linear form L 1 , . . . , L d involves coefficients of degree ≥ k. For i = 0, . . . , k -1, we denote as before by ρ(i) the linear map

Given solutions U 1 , . . . , U d , we consider a finitely generated vector space or module V containing U 1 , . . . , U d and the k-kernel of A 1 , . . . , A d . We have then

where

is the linear form obtained from L j by considering the coefficients of z j . The power series ρ(i)U j ∈ V is thus a linear combination of U 1 , . . . , U 1 and of the k-kernel of A j . The set V contains thus the k-kernel of U j and U 1 , . . . , U d are all k-regular.

If there are linear forms among L 1 , . . . , L d which are of degree ≥ k, we introduce the k-regular series A d+1 = zA 1 , . . . , A 2d = zA d , the series U d+1 = zU 1 , . . . , U 2d = zU d , the linear forms L d+1 = zL 1 , . . . , L 2d = zL d ∈ zR[z][x 1 , . . . , x d ] and set α d+1 = . . . = α 2d = 0. We have then the equations U i (z) = A i (z) + zL i (U 1 (z k ), . . . , U n (z k )) and the identities α i = U i (0) for i = 1, . . . , 2d. Modifying a linear form L j of degree ≥ k by substituting all occurences of z k+i U j (z k ) with z i U d+j (z k )


Denis Darpy
email: denis.darpy@dauphine.fr

Isabelle Prim-Allaz

Potential effects of psychological reactant consumers on relationships marketing programmes

Keywords: Psychological reactance, relationship marketing, long term commitment, relationship orientation . Potential effects of psychological reactant consumers on relationships marketing programmes

This conceptual article presents the current criticisms to relationship marketing and important concepts developed in the psychology to understand consumer reactance toward services packaged as formal contracts. A first part presents the numerous questions that researches are raising about the willingness of people to commit to services. A second part shows that psychological reactance, already studied in the context of non product choice, is an interesting concept to revisit the relationship marketing paradigm: are all customers willing to commit to a service or are they forced against their own freedom? The last section develops an integrative conceptual framework of the different concepts introduced in this research.

Introduction

The large financial investments required to develop new telecommunication technologies or efficient transportation systems are often justified through the return on investment during the life of a customer. To make sure that the retention of the customer is as long as possible, the new transportation and information services are most of the time available to clients through subscriptions which imply some kind of commitment and relationship between a supplier/provider and a customer. While the consumer is facing a growing number of commercial demands, he may develop psychological reactance (PR). As research and practices are moving from a transactional approach to a relational paradigm, psychological reactance, defined as the reaction of people against attempts to constrain their free behaviour [START_REF] Brehm | A theory of Psychological Reactance[END_REF], may be an explanation of non subscription of services for some clients. Indeed, relationship marketing makes the assumption that consumers agree on the relational contract while it is not obvious that all customers are willing to drop some of their freedom of choice to enter in these relationships [START_REF] Barnes | Closeness, Strength, and Satisfaction: Examining the Nature of Relationships between Providers of Financial Services and their Retail Customers[END_REF]. Even though the marketer is always offering more value to customers, he puts more pressure at the same time on the same customer [START_REF] Dussart | Licencier les clients[END_REF]. This pressure is often materialised through a unidirectional commitment of the client towards the company. Should the client / company relation be a formal wedding ? However, the customer is now more and more aware of the marketing techniques and hence becomes more reluctant to business offers.

Moreover, relationship marketing works and practices are scarcely taking the lost of the client freedom as a cost for him until recent works develop the study of the cost of retention of subscribers [START_REF] Gustafson | The Effects of Customer Satisfaction, Relationship Commitment Dimensions, and Triggers on Customer Retention[END_REF]. This conceptual article presents the current criticisms to relationship marketing and important concepts developed in the psychological field research to understand the attitude of consumer reactance toward services packaged as formal contracts. The field of services has been chosen for this study as the characteristics of services (e.g., intangibility, variability, heterogeneity…) reinforce the willingness of service companies to create and develop durable and deep relationships and commitment with their customers.

The paper is developed in three parts. A first section presents the numerous questions that researchers are raising about the willingness of people to commit to services. A second part will show that psychological reactance, as developed by psychologists and consumer researchers in a product transaction context, is an interesting concept to revisit the relationships marketing paradigm: are all customers willing to commit to a service or are they forced against their own freedom? The last section of the article builds an integrative conceptual framework of the different concepts introduced in this research. [START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF] have defined relationship marketing as "establishing, developing and maintaining successful relational exchanges". Relationship marketing has also been defined more precisely as an "effort to identify, build and sustain a network of individual consumers, and the continuous strengthening of this network in the advantage of both parties by means of interactive, personal, and value-adding contacts during a long period" (Shani and Chalasani, 1992, p. 44). The notions of networking, interactivity and success are central to relationship marketing. Indeed, relationship marketing is theoretically built upon three main components: long term commitment, mutual knowledge win-win partnership [START_REF] Grönroos | Service Management and Marketing[END_REF].

Relationship marketing development and limits

Components of relationship marketing

Long term commitment is key to networking, and a condition to establish reliable partnerships. Customers are aware of it, even though it is not always accepted. Commitment can be defined as a "buyer's enduring desire to continue a relationship with a seller accompanied by his willingness to make efforts at maintaining it" [START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF].

Authors assume a positive link between relationship satisfaction and commitment (e.g., [START_REF] Bolton | A Dynamic Model of the Duration of the Customer's Relationship with a Continuous Service Provider: the Role of satisfaction[END_REF][START_REF] Ganesan | Negotiation Strategies and the Nature of Channel Relationships[END_REF]. [START_REF] Ganesan | Negotiation Strategies and the Nature of Channel Relationships[END_REF] finds strong empirical support for the path from satisfaction to long-term orientation. Moorman, Zaltman, and Deshpandé (1992) suggested that buyers who are committed to a relationship might have a greater propensity to act because of their need to remain consistent with their commitment to avoid cognitive dissonance. Commitment is complex; it is viewed as a multidimensional construct. A generally accepted approach in the literature presents "commitment" as being twodimensional: (1) affective commitment, emotion based feeling which involves a certain degree of reciprocity and (2) calculative commitment, a colder and more rational economicbased relation (Gustafsson, Johnson & Roos, 2005). It reflects both emotional and cognitive consumer decision processes. Similarly, [START_REF] Gutiérrez | The Consumer's Relational Commitment : Main Dimensions and Antecedents[END_REF] propose three aspects to describe commitment: one referring to the true current behaviour, promises and sacrificebehavioural dimension-, another that contemplates desires and feelings -affective dimensionand a third one that indicates the intention of future commitment. Customer loyalty is then presented as a major consequence of relational policies based on trust and commitment [START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF][START_REF] Garbarino | The Different Roles of Satisfaction, Trust and Commitment in Customer Relationships[END_REF].

While "commitment" seems natural to customers engaged in relationships marketing, the two other characteristics of Relationship Marketing, mutual knowledge and win-win partnership, may be challenged. Indeed, a commercial relationship is most of the time deeply asymmetrical. The company is knowing much more of the customers, with the help of databases and data mining research methods, than the customers of the company. Moreover, customers often have the feeling that companies mostly work on making profits rather than taking their interests into account. Wouldn't be legitimate for consumers to doubt about the sincerity of the firm empathy?

The emerging field research on consumer resistance and ethics, On the basis of the micro power developed by Foucault, marketing tools and practices are described as disciplinary processes in order to develop a social control on customers (Marsden, 2001[START_REF] Roux | La résistance du consommateur : Conceptualisations, Echelles d'observation et proposition d'un cadre d'analyse[END_REF]: information and intelligence techniques are allowing companies to better know their customers, segmentation techniques allow categorizations, communication techniques influence customers… When these customers get aware of the existence of these information and manipulation, they may become suspicious and reactant. The continuous merger of CRM and data mining techniques is raising concerns among consumers. The necessity of long term commitment of the client, the deficient win-win partnership, as the doubts of customers about the overt behaviour of firms, are sources of limits to the development of relationships marketing.

Relationship marketing traditional approach requires conditions which are rarely available.

Hence, consumers may doubt of the sincerity of the company to develop links rather than just selling goods and services.

Customer own interest in relationship marketing

Scholars have developed relationship marketing with the interest of the company as primary objective. When the customer's interest is taken as the research focus, the perspective may change. Fournier and al. (1998, p. 44) have outlined a major limit to relationship marketing: "we haven't looked close enough to see that the consumer is not necessarily a willing participant in our relationship mission". There is a need to understand the choice that companies make between transactional and relational marketing. Further research should investigate how marketing practices reflect perceived customer need structures and customers' preferences for transactional and/or relational exchange [START_REF] Coviello | How Firms Relate to their Markets: an Empirical Examination of Contemporary Marketing Practices[END_REF].

As increased marketing actions raise the number of solicitations toward customers, some fellows have worked on the assumption that customers may have different levels of relationship orientation. [START_REF] Lovelock | Classifying Services to Gain Strategic Insights[END_REF] and [START_REF] Oldano | Relationship segmentation : enhancing the service provider/client connection[END_REF] propose to segment the market according to desired type of business relationship. [START_REF] Barnes | Closeness, Strength, and Satisfaction: Examining the Nature of Relationships between Providers of Financial Services and their Retail Customers[END_REF] makes the distinction between two types of customers: those who desire a deep and warm relationship and those who are looking for a more distant relationship. This concept is also called client's relational predisposition and defined as an enduring tendency of the customer to expect and value a relational approach from its service provider. A relational approach is viewed as: a bilateral relationship; a long term commitment; a personalized and tailored-made service [START_REF] Bahia | Les conséquences de la prédisposition relationnelle du client[END_REF].

According to their level of relational predisposition, customers are more or less willing to accept one or another type of business exchange (relational or transactional) [START_REF] Benamour | Transactional versus Relational Customer Orientation: Developing a Segmentation Tool in the French Banking Industry: An Exploratory Study[END_REF].

Relative acceptation by the customer of relational and transactional exchange

Although the relevance of transactional and relational marketing variables seems obvious, knowledge about the conditions of their use is still limited [START_REF] Jackson | Winning and Keeping Industrial Customers : The Dynamics of Customer Relationships[END_REF][START_REF] Perrien | Le marketing relationnel: oui[END_REF][START_REF] Fruchter | Transactions vs. Relationships: What should the Company Emphasize?[END_REF]. "The challenge for academics and managers, then, is to make sound predictions of the effectiveness of the two types of marketing activities for different products, customers and market conditions" [START_REF] Fruchter | Transactions vs. Relationships: What should the Company Emphasize?[END_REF].

Authors study the relational/ transactional customer orientation along three different approaches: (1) The link between the customer personality and the global attitude towards the relationship [START_REF] Perrien | The Meaning of Relationship Marketing: A Pilot Study[END_REF][START_REF] Perrien | Les facteurs explicatifs et normatifs d'une approche relationnelle: La perception des clients organisationnels[END_REF]; [START_REF] Barnes | Closeness, Strength, and Satisfaction: Examining the Nature of Relationships between Providers of Financial Services and their Retail Customers[END_REF] with a variable such as consumer relationship proneness; (2) the consequences of the relationship such as psychological, social, and economic benefits [START_REF] Barnes | Closeness, Strength, and Satisfaction: Examining the Nature of Relationships between Providers of Financial Services and their Retail Customers[END_REF][START_REF] Gwinner | Relational Benefits in Services Industries: The Customer's Perspective[END_REF]; (3) the contextual elements such as service characteristics -importance, involvement, quality, complexity- [START_REF] Berry | Relationship Marketing of Services -Growing Interest, Emerging Perspectives[END_REF][START_REF] Benamour | Transactional versus Relational Customer Orientation: Developing a Segmentation Tool in the French Banking Industry: An Exploratory Study[END_REF].

The "consumer relationship proneness" (CRP), as a mediating variable between the impact of need for social affiliation and behavioural intentions [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF], opens new directions to study the impact of personality on relationship marketing. For these authors, CRP represents a personality trait that reflects a consumer's relatively stable and conscious tendency to engage in relationships with sellers of a particular product category. From a service encounter perspective, CRP can be expected to play a crucial role in a people-based service encounters (like hairdressers or dentists). In line with the literature on interpersonal relationship they define the need for social affiliation (NSA) as a preference to be with other people and to engage in relationships. People with a high need for social affiliation do not look for social rewards, but are rather intrinsically valuing the relationships with other people.

From a commercial friendship perspective, the concept of NSA has been referred to as sociability [START_REF] Price | Commercial Friendships: Service Provider-Client Relationships in Context[END_REF] and can be considered as "a tendency to affiliate with others and to prefer being with others to remaining alone". [START_REF] Price | Commercial Friendships: Service Provider-Client Relationships in Context[END_REF] regard sociability as one of the factors potentially contributing to the formation of commercial friendship in a services setting. [START_REF] Forman | The Depersonalization of Retailing: its Impact on the "lonely consumer[END_REF] claimed that people in search for human contact are willing to engage in long-term relationships. Development of postmodern tribes and brand communities, sometimes out of the control of the brands, represents an emerging type of relationships between consumers [START_REF] Cova | Community and consumption[END_REF]Muniz and O'Guin, 2001) [START_REF] Gwinner | Relational Benefits in Services Industries: The Customer's Perspective[END_REF] argue that customers look for different types of benefits in a business exchanges and that according to which type of benefits they value, they will prefer transactions or relational exchanges. Findings from two studies across three categories of services indicate that consumer relational benefits can be categorized into three distinct types of benefits: (1) social benefits: beyond the benefits of the basic service, customers search for fraternization and likable relationships. Therefore, they are looking for certain personal recognition; (2) psychological benefits: they are often linked with comfort, feelings of security and particularly trust or confidence in the provider; (3) customization benefits: the final category of relational benefits relates to customization of the offered service. According to which type of benefits they value customers can be categorized. Another attempt to use this approach was proposed by Prim-Allaz and [START_REF]L'orientation relationnelle des clients : un nouvel outil de segmentation ?[END_REF]. These authors demonstrate that in the French banking and medical contexts, it is possible to identify two main categories of customers: those who only value social benefits and those who value both social and economic benefits.

Finally, customer relationship orientation is also presented as being linked to the situation. [START_REF] Benamour | Transactional versus Relational Customer Orientation: Developing a Segmentation Tool in the French Banking Industry: An Exploratory Study[END_REF] believe that the same customer may have different level of relationship orientation according to its implication in the buying situation or the level of perceived risk. This concept is also called client's relational predisposition and defined as an enduring tendency of the customer to expect and value a relational approach from its service provider [START_REF] Bahia | Les conséquences de la prédisposition relationnelle du client[END_REF].

Reactance to long term commitment

Psychological reactance

As mentioned in the first paragraph of this paper, relationship marketing is widely based on commitment [START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF]. This commitment can be either contractual or not. In the first case, the formalization of the commitment may be seen by customers as a formal and indefeasible tie. This tie may then be interpreted as a threat to the customer freedom of choice as for some promotional influence, manipulative advertisement, ... [START_REF] Clee | Consumer Behavior and Psychological Reactance[END_REF].

The contractualization may be seen by some customers as a threat to their freedom for future choices. As a consequence, some customers develop strategies to reply to this feeling of freedom reduction through a commitment refusal. Actually, customers may have different cognitive, affective and behavioural consequences for different levels of perceived decision freedom. [START_REF] Lessne | Reactance Theory in Consumer Research : the Past, Present and Future[END_REF] suggest that Psychological Reactance should be applicable to a limited set of conditions where consumers'freedom of choice is threatened. They also suggest "that the theory [should] not be tested in the strict theory testing perspective but rather that the theory [should] be used as a guiding framework in investigations of important phenomena".

Psychological reactance (PR) is "the motivational state that is hypothesized to occur when a freedom is eliminated or threatened with elimination" (Brehm & Brehm, 1981, p. 37). The theory indicates that when a perceived freedom is eliminated or threatened with elimination, the individual will be motivated to re-establish that freedom. Given that an individual perceives a specific freedom, any force on the individual that makes it more difficult for him or her to exercise that freedom constitutes a threat [START_REF] Brehm | A theory of Psychological Reactance[END_REF][START_REF] Brehm | Psychological Reactance[END_REF].

Two major antecedents to Psychological Reactance were identified [START_REF] Brehm | A theory of Psychological Reactance[END_REF][START_REF] Wicklund | Freedom and Reactance[END_REF] : the strength to freedom and trait reactance freedom. The first one is situational : "the number of equally valued alternatives is the most important antecedent of perceived decision freedom" [START_REF] Walton | Consumer Decision Making and Perceived Decision Freedom[END_REF]. The second one recognizes that individuals may vary in their trait proneness to reactance arousal [START_REF] Shen | Psychometric properties of the Hong psychological reactance scale[END_REF]. Indeed Psychological Reactance is associated with defensiveness, dominance and aggressiveness [START_REF] Dowd | Motivational Components of Psychological Reactance[END_REF]. Reactant people have a tendency to act without considering potential consequences [START_REF] Buboltz | Personality and Psychological Reactance: Extending the Nomological Net[END_REF].

There are three types of threats to freedom: (1) interpersonal or social threats; (2) impersonal1 threats and (3) self-imposed threats. In research as the present one, the focus is put on interpersonal and social threats.

In persuasion models, PR is presented as a mediator between communication and attitude/behaviour [START_REF] Fitzsimmons | Reactance to Recommendations: When Unsolicited Advice Yields to Contrary Responses[END_REF]Dillard & Shen, 2005). Direct restoration of freedom involves doing the forbidden act. In addition, freedom may be restored indirectly by increasing liking for threatened choice, derogating the source of threat, denying the existence of threat or by exercising a different freedom to gain feeling of control and choice (Dillard & Shen, 2005).

Persuasive attempts of all sorts, including public health campaigns, often fail to produce the desired effect. In some cases, they even produce results directly at odds with their intent. The theory of PR provides one theoretical perspective through which these miscarriages might be understood. The theory contends that any persuasive message may arouse a motivation to reject the advocacy. That motivation is called reactance (Dillard & Shen, 2005). From this inception to the present, the theory may be called upon to explain resistance to long-term commitment.

One of the reason why there has been so little empirical research on PR in the context of consumer behaviour may be that the theory is not seen as having managerial ramifications [START_REF] Lessne | Reactance Theory in Consumer Research : the Past, Present and Future[END_REF]. According to us, this point of view was relevant in a transactional perspective but is no longer defensible in a relational perspective which implies customers' long term commitment. Psychological Reactance should take place among other concepts to predict the propensity to sign a long term contract. Table 1 is summarising the different personality related concepts developed to explain the lack of direct link between satisfaction and commitment.

Table 1 -Personality related concepts to commitment Definition Need for social affiliation Preference to be with other people and to engage in relationships [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF] Opportunism Opportunistic behaviours araise when the consumer makes sure to stay aware of competitive promotional offers in order to take advantage of them, whatever the link he may have with the current service provider [START_REF] N'goala | Proposition d'une conceptualisation et d'une mesure relationnelle de la fidélité[END_REF].

Customer relationship orientation or relational predisposition

Enduring tendency of the customer to expect and value a relational approach from its service provider [START_REF] Bahia | Les conséquences de la prédisposition relationnelle du client[END_REF]. There are different approaches to relationship orientation: according to the benefits the customer values; according to contextual elements and according to personality traits (cf. infra, customer relationship proneness) -Benamour & Prim, 1999-.

Customer relationship proneness

Personality trait that reflects a consumer's relatively stable and conscious tendency to engage in relationships with sellers of a particular product category [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF]

Commitment

Buyer's enduring desire to continue a relationship with a seller accompanied by his willingness to make efforts at maintaining it [START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF]. It implies the adoption of a long term orientation toward the relationship -a willingness to make short-term sacrifices to realize long-term benefits from the relationship (Dwyer, Schurr & Oh, 1987;[START_REF] Weitz | The Use of Pledges to Build and Sustain Commitment in Distribution Channels[END_REF]. Commitment is a motivational phenomenon [START_REF] Wiener | Commitment in Organizations. A Normative View[END_REF] Commitment is generally considered as multi dimensional.

Affective commitment Degree to which the membership is psychologically bonded to the organization on the basis of how favourable it feels about the organization [START_REF] Gruen | Relationship Marketing Activites, Commitment, and membership Behaviors in Professional Associations[END_REF] Calculative commitment Degree to which a membership is psychologically bonded to the organization on the basis of the perceived costs (economic, social, and status related) associated with leaving the organization, based on the self-interest stake in a relationship [START_REF] Gruen | Relationship Marketing Activites, Commitment, and membership Behaviors in Professional Associations[END_REF].

Psychological reactance

Reaction of people against attempts to constrain their free behaviour [START_REF] Brehm | A theory of Psychological Reactance[END_REF]

Integrative conceptual framework

Based on this literature review, we suggest to develop a set of hypothesis and propose a theoretical model. The first aim of this research is to explain why some customers accept to engage in long term contracts while others are reactant. Contracts have this specificity that they are not reducing present choices (customer may or may not contract) but future choices (as soon as the contract is signed, they may have reduced choices).

As seen in the literature, we assume that there is a positive relationship between the level of commitment and positive behaviours toward a long-term relationship (Gutièrrez et al, 2004;[START_REF] Gruen | The Outcome Set of Relationship Marketing in Consumer Markets[END_REF][START_REF] Morgan | The Commitment-Trust Theory of Relationship Marketing[END_REF]. The level of commitment may be the consequence customer relationship proneness [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF].

Customer relationship proneness (CRP) is the concept chosen to represent the relationship orientation. CRP is a consequence of both customer opportunism -negative link-(Prim-Allaz & [START_REF]L'orientation relationnelle des clients : un nouvel outil de segmentation ?[END_REF] and need for social affiliation -positive link- [START_REF] Bloemer | The Impact of Need for Social Affiliation and Consumer Relationship Proneness on Behavioural Intentions: An Empirical Study in a Hairdresser's context[END_REF]. More the customer relationship proneness is high, more the affective commitment is increasing. The psychological reactance (PR) is an antecedent of Customer Relationship Proneness and of calculative commitment. When psychological reactance is high, customer relationship proneness level will decrease and as a consequence will weaken the importance of affective commitment. However, psychological reactant people may select to sign a long term contract when the calculated commitment is strong enough. Reactant people may favour quality and price attributes instead of relationship meanings. Psychological reactance influences calculated and affective commitment through a trade-off evaluation with CRP.

The propensity to sign a long term contract (such as subscriptions to an internet provider, to a phone company or with a health doctor) will be determined either for calculated or for affective reasons. Two different types of commitment may appear which will create variance between the individuals. A unique loyalty program may not be suitable to both segments of people. As suggested by [START_REF] Roux | La résistance du consommateur : Conceptualisations, Echelles d'observation et proposition d'un cadre d'analyse[END_REF] the link between non loyalty and reactance will have to be investigated. The recent works on brand communities should also be revisited through the psychological reactance framework. Indeed, psychological reactant people may appear to be reluctant to adhere to clubs and communities, for the same reason they reject bindings in contracts. However brand communities may attract reactant people when these postmodern tribes don't develop relationships with the parent brands.

Conclusion

To answer to the current question "are all customers willing to commit to a service or are they forced against their own freedom?", the use of psychological concepts is proposed.

Psychological reactance seems to offer very interesting research avenues in understanding the differences of commitment between customers in long term relationships. This paper ends with the proposition of an integrative conceptual framework based on five main concepts: psychological reactance; customer relationship proneness; calculated and affective

Figure

  Figure 1 -Theoretical model

Interpersonal threats involve influence attempts while impersonal don't.