Jose Ignacio

Aguilar Río

Jose Aguilar
email: jose.aguilarrio@univ-paris3.fr

Cédric Brudermann

Dagmar Abendroth-Timmer

Resorting to tandem learning in academic language teacher training programmes: evidence from the literature of / and the field

Keywords: blended learning, pedagogical engineering, pre-service language teacher training, mediation, sociocultural theory, tandem learning

This paper explores the potential and limitations of tandem learning to foster professional development amongst pre-service language (L2) teachers, both from a theoretical and a practical point of view. As such, a review of the literature of the field is first proposed. It suggests that while "tandem learning" is an evenly distributed and rather commonly used pedagogical practice to promote L2 proficiency development and intercultural awareness among learners, this instructional approach paradoxically seems to be unusually relied upon in the professional field to foster skill development. This review then leads to an empirical part in which three tandem and collaborative pre-service teacher training projects are described and analyzed. This part notably focuses on the pedagogical and professional benefits / weaknesses of such an approach to train would-be L2 teachers. We conclude by showing that, while evidence from the literature of / and the field reveals that tandem learning may be a sustainable pedagogical option to consider to work towards achieving professional objectives, the recognition of the potential of this very instructional practice to foster skill development seems more problematic for various actors pertaining to the academic sphere.

Tandem learning: pedagogical innovation as a recycling endeavour

According to Tricot (2017, 8) "pedagogical innovation" may be fathomed as an everrenewing recycling process of sorts, by virtue of which any major technological breakthrough (such as the advent of the printing press or the Internet, for instance) is liable to upgrade what then become "past" pedagogical outcomes. This definition places the principle of innovation at the level of the individual rather than at the level of a given community of practice [START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF]. And, following Tricot, tandem learning does not stricto sensu represent an innovation as such a pedagogical initiative has indeed been used for a long time 1 as a possible materialization of autonomous (extra-institutional) language learning practices that involve two partners with different / reciprocal linguistic backgrounds, with the underlying idea that this mutually supportive partnership would help them make progress (Bechtel 2003, 15). What may be seen as innovative however in tandem learning initiatives is that the latest technological developments -namely the dematerialization and (a)synchronicity of communications that the Internet has brought -have allowed for the deconstruction of this, at least, oneon-one interactional dynamics (Funk, Gerlach, and Spaniel-Weise 2017). And onefundamentally pedagogical -possible result of such a recycling process is a diversification of the ways in which mediation can be carried out in the digital era (Brudermann et al. in press) to reach specific objectives. For example, at a time when 1 [START_REF] Comenius | La grande didactique ou l'art universel de tout enseigner à tous[END_REF] peer-teaching practices and [START_REF] Freinet | Les Dits de Mathieu: une pédagogie moderne de bon sens[END_REF] letter exchange programmes may be regarded as proto-versions of contemporary tandem learning practices.

the "respect for cultural otherness" (OECD 2016) is listed as a capital skill -to which learners, pupils and students from all over the world must be sensitized -tandem learning seems to appear as a second to none mode of delivery worth being considered as it allows to (i) be in direct contact with "culturally-different" communities and (ii)

keep pace with a more and more global, integrated and multicultural society.

However, in state-run educational settings, to have learners make the most of their language practices with other individuals across the world, their language exchanges need to be monitored by staff members. This state of affairs therefore implies for the teachers to be competent practitioners to accompany such collaborative experiences and that they were consequently trained -and, logically, in tandem language learning environments -to gain the necessary skills to do so in the first place.

Yet, the literature of the field regarding the use -and, by cascading effect, the potential -of tandem learning to foster the development of professional skills seems paradoxically scarce. To bridge this gap, what follows is an exploration -both from a theoretical and a practical point of view -of the potential and limitations of this mode of delivery to foster professional development amongst pre-service language (L2) teachers.

Tandem learning through time and space: reviewing the literature of the field

In order to shed light on the links between tandem learning and the development of professional competence in today's socio-educative context, we conducted a literature review of the field. To do so, four databases were arbitrarily chosen and searched:

Dialnet, EBSCohost, HAL and ProQuest. The research queries which were made in English, French and Spanish in the databases were the following: a) "tandem AND teacher training OR teacher education OR professional development AND language teaching"; b) "tandem AND enseignant (teacher) AND formation (training) OR langue"; c) "tandem learning AND language teacher training". The time period which was defined for this survey spanned about four decades (i.e. from 1980 to 2017). In order to keep a formal coherence among the data contained in the references, we purposely chose to only retain the results which matched the two following criteria: first, the selected articles had to contain both an abstract (proposed by either the authors or the editor) and a list of keywords. Secondly, in order to deal with studies which had an explicit educational scope, the selected articles had to be published in peer-reviewed journals specialising in applied linguistics or the educational sciences 2 . The search which was conducted allowed to form a corpus of 43 references which we further assembled within a Zotero collection 3 .

Quantitative analysis of the corpus

A quantitative analysis of the corpus was first carried out. The "keywords in context" function of Voyant Tools 4 was used to parse through the 43 abstracts of the corpus. This statistical search allowed to identify two recurring thematic trends which appeared as peripheral and / or central to the relationship between "tandem-learning" and "language teacher training", namely teachers as study subjects and subjects dealt with. These trends were further treated as three distinct categories. In what follows, each category will be presented and discussed in a dedicated section.

2

The filters "subject" and "journal title"-available in the EBSCohost, HAL and ProQuest databases -proved useful to satisfy this criterion. The first unfiltered queries yielded results in disciplines other than applied linguistics and the educational sciences -namely demographic studies, economics, marketing, political sciences and psychology.

3

Cf. https://tinyURL.com/ybz8qtf4 (last accessed on October, 14th 2017).

Teachers as study subjects

In the corpus, 19 references feature teachers as study co-participants whose functions vary depending on the scopes / educational contexts of the articles. The central or peripheral nature of teachers as subjects of study is presented in table 1.

Categories of "teachers" dealt with in the articles Occurrences In-service teachers (whatever the subject) 7 Pre-service teachers (whatever the subject) 5

Both teachers and students (whatever the level and subject) 4

Teachers and researchers 2

Teacher trainers 1

Total number of references 19

Table 1. Teachers as study subjects: evidence from the corpus.

As table 1 above suggests, a diversity of target actors revolves around the socioprofessional category "teacher". The co-occurrence of two profiles of actors in some of the references of the corpus suggests forms of collaborative work, as is the case with "teachers and researchers" and "teachers and students". The status of these actors, as well as the hierarchical dynamics among them in some of the references of the corpus, is discussed in what follows.

Subjects dealt with by the collaborative groupings highlighted in the corpus

Among the 43 references of the corpus, 36 explicitly present distinct learning objectives.

Objectives Occurrences

Development of teaching skills 19

Behaviourally oriented objectives 7

Language related objectives

Total number of references 36

Table 2. Objectives of the studies dealt with in the corpus.

For the first category (development of teaching skills), "pedagogical approaches" [START_REF] Tanghe | Build[] Something Which Alone We Could Not Have Done': International Collaborative Teaching and Learning in Language Teacher Education[END_REF], "assessment design" [START_REF] Bennett | From Cognitive-Domain Theory to Assessment Practice[END_REF], "telecollaborative competences" (O'Dowd 2015), "intra-/ inter-personal dimensions of critically reflective practice" [START_REF] Skattebol | Affect: A Tool to Support Pedagogical Change[END_REF], "pedagogical knowledge" [START_REF] Karimi | Scaffolding Teacher Cognition: Changes in Novice L2 Teachers' Pedagogical Knowledge Base through Expert Mentoring Initiatives[END_REF] and "trainee teachers' language and thinking skills" [START_REF] Taylor | Thinking, Language and Learning in Initial Teacher Education[END_REF]) are some of the specific teaching skills addressed by the studies included in the corpus. These skills have to do with explicit and direct pedagogical practices which seek to develop them, either as training objectives or as instances of teaching practices aiming at triggering the experienced teachers' critical reflexion.

The category "behaviourally oriented" comprises seven studies which present training objectives in the form of expected attitudes or behaviours, such as "emotion", among master's French language trainee teachers [START_REF] Xue | Interaction En Formation Des Enseignants de Langue : Sensibilisation de Futurs Enseignants Sur Le Plan Émotionnel Par Un Dispositif Hybride Interactif[END_REF], "actitud colaborativa" (collaborative attitude), between undergraduate would-be primary school teachers [START_REF] Ferrer | La colaboración en eportafolios con herramientas de la Web 2.0 en la formación docente inicial[END_REF], rural school teachers' "resilience" in Africa's most southern areas [START_REF] Ebersöhn | Rurality and Resilience in Education: Place-Based Partnerships and Agency to Moderate Time and Space Constraints[END_REF] Following the quantitative analysis of the corpus, a qualitative survey was conducted. In what follows, a presentation of this qualitative analysis is proposed.

Qualitative analysis of the corpus

In the previous section, a general, quantitative analysis of the corpus was presented. In this section, the focus is on a more detailed scrutiny of the aims and objectives of the studies accounted for in the corpus. For this survey, out of the 43 references, 9 turned out to deal with explicit tandem experiences / collaborative work environments which had educational aims (whatever the subject or level of education). These 9 references were hence retained to form a sub-corpus 5 , which we further used to conduct the qualitative analysis. To do so, the corpus was first manually coded. Then, tagging the corpus further led to assign the data into two main categories, namely "tandem implementation as a pedagogical practice" and "problems inherent to tandem implementation and possible solutions". As table 3 shows, the collaborative work experiences accounted for in the subcorpus took the form of either explicit tandems [START_REF] Salomã | Collaborative Language Learning in Teletandem: A Resource for Pre-Service Teacher Education[END_REF][START_REF] Siebold | El aprendizaje intercultural en un curso de Tándem[END_REF][START_REF] Tanghe | Build[] Something Which Alone We Could Not Have Done': International Collaborative Teaching and Learning in Language Teacher Education[END_REF] or dyads [START_REF] Karimi | Scaffolding Teacher Cognition: Changes in Novice L2 Teachers' Pedagogical Knowledge Base through Expert Mentoring Initiatives[END_REF]. Among these, two were concerned with explicit linguistic learning objectives [START_REF] Salomã | Collaborative Language Learning in Teletandem: A Resource for Pre-Service Teacher Education[END_REF][START_REF] Siebold | El aprendizaje intercultural en un curso de Tándem[END_REF]. Other forms of collaborative work included workgroups of up to 5 members [START_REF] Gascón | Aprendizaje cooperativo y evaluación formativa en marketing: tándem ganador en docencia universitaria[END_REF][START_REF] Nutley | The Many Forms of Research-Informed Practice: A Framework for Mapping Diversity[END_REF][START_REF] Xue | Interaction En Formation Des Enseignants de Langue : Sensibilisation de Futurs Enseignants Sur Le Plan Émotionnel Par Un Dispositif Hybride Interactif[END_REF], double tandems / dyads [START_REF] Guichon | Training Future Language Teachers to Develop Online Tutors' Competence through Reflective Analysis[END_REF]) and groups whose precise working modalities were not specified [START_REF] Lavy | Who Benefits from Group Work in Higher Education? An Attachment Theory Perspective[END_REF]. 5 references accounted for specific, language teacher training programmes, including trainee teachers, pre-service, inservice and teacher educators [START_REF] Guichon | Training Future Language Teachers to Develop Online Tutors' Competence through Reflective Analysis[END_REF][START_REF] Karimi | Scaffolding Teacher Cognition: Changes in Novice L2 Teachers' Pedagogical Knowledge Base through Expert Mentoring Initiatives[END_REF][START_REF] Salomã | Collaborative Language Learning in Teletandem: A Resource for Pre-Service Teacher Education[END_REF][START_REF] Tanghe | Build[] Something Which Alone We Could Not Have Done': International Collaborative Teaching and Learning in Language Teacher Education[END_REF][START_REF] Xue | Interaction En Formation Des Enseignants de Langue : Sensibilisation de Futurs Enseignants Sur Le Plan Émotionnel Par Un Dispositif Hybride Interactif[END_REF]. In the following sections, the focus will be on analysing how these five references address intercultural communication and exchange via electronic devices and access to the web -i.e. tandem learning -as a pedagogical and organisational element.

Tandem implementation as a pedagogical practice

The qualitative analysis of the sub-corpus revealed that "social cohesion" was strongly related to tandem learning and that this parameter was also positively regarded in the academic sphere to foster skill development. According to the analysis, social cohesion includes both inter-personal -such as the encouragement of mediation (Salomã 2011, 154) -and intra-personal aspects (for instance, reflexivity (Tanghe and Park 2016, 6) or perceived psychological detachment regarding (un)conscious beliefs (Tanghe and Park 2016, 9;Xue and Schneider 2015, 14)). The analysis also seems to point towards the relevance of tandem learning to both have trainee teachers develop their professional awareness [START_REF] Guichon | Training Future Language Teachers to Develop Online Tutors' Competence through Reflective Analysis[END_REF][START_REF] Karimi | Scaffolding Teacher Cognition: Changes in Novice L2 Teachers' Pedagogical Knowledge Base through Expert Mentoring Initiatives[END_REF][START_REF] Salomã | Collaborative Language Learning in Teletandem: A Resource for Pre-Service Teacher Education[END_REF][START_REF] Tanghe | Build[] Something Which Alone We Could Not Have Done': International Collaborative Teaching and Learning in Language Teacher Education[END_REF] and lead international students to enhance their sense of belonging to communities that they may otherwise have perceived as remote from their habits and comfort zones [START_REF] Siebold | El aprendizaje intercultural en un curso de Tándem[END_REF].

Problems deriving from the implementation of tandem learning initiatives

The analysis also highlighted that the implementation of tandem learning initiatives came with limitations. These seem indeed to be related to the participants' attitude and roles in tandem learning settings, the way the tandem learning work environments were designed, or the apparent (ir)relevance of the research methods used by the researchers during the experiments to pursue the objectives they had set.

For instance, Siebold and Zulategui (2012, 505) discuss the unpredictability of the durability of tandem learning initiatives. These authors suggest that even when an initial, apparent, enthusiasm among the members of a tandem experience is perceived (particularly during the early stages), this positive attitude tends to quickly fade away and to further lead to their later disengagement. According to the authors, one possible way to circumvent this difficulty is to define a clear, institutional frame, which sustainably engages the participants. In a similar way, Guichon (2009, 180) highlights the potential pitfall of unequal commitment from members of a same dyad to perform collective tasks, which the author explains by a diverging capacity for self-introspection among the trainee teachers. Gascón and De-Juan-Vigaray (2015, 375) also underscore a certain unpredictability regarding the amount of work which may (or not) be sustained by the tandem learning / collaborative work dynamics. The authors suggest that having the participants of a tandem or group to collaboratively identify one another's strengths may help them attribute roles and functions within the tandem / group in a more appropriate way and, in doing so, improve the work dynamics. Also, the authors insist on the likelihood that using some form of random appointment method to form groups may engender the participants' surprise or resistance. This question is also addressed by Finally, as regards research endeavours, Guichon (2009, 180) and Karimi and Nouruzi (2017, 46) insist on the limits resulting from the selected data collection and analysis methods, since these prevented from clearly measuring possible correlations between the implemented tandem learning initiatives and the expected outcomes.

Discussion

The quantitative and qualitative analyses we conducted sought to better identify what the literature of the field considered as benefits and limitations when resorting to tandem learning to foster skill development. The findings we brought forward in the previous sections have extensively dealt with these aspects. In particular, they suggest that tandem learning is a rather evenly distributed pedagogical practice -whatever the educational setting (primary, secondary and higher education), subject or project promoter -but that this instructional approach is rather unusually relied upon in the professional field to foster skill development. However, these findings should be regarded with caution: the corpus we used was gathered according to the arbitrary criteria presented above. Besides, the presented qualitative analysis was based on a subcorpus made up of nine references. The term "analysis" may therefore not be totally appropriate to refer to what was proposed, for the benefit, for example, of a somehow less specific notion such as "trend". Nonetheless, whatever the approaches chosen to collect and analyse a corpus of references dealing with the potential / limitations of tandem learning to foster skill development, the fact is that the present authors went through a major difficulty when trying to identify pieces of academic writing dealing with this topic. And trying to outline "trends" rather than bringing "proper" analyses to the fore to explore this relationship is an indicator that the literature of the field on the topic is scarce. In order to complete this approach and delve deeper into the exploration of the potential / limitations of tandem learning to foster skill development, cross referencing the above-mentioned findings with other data, taken from the field, could also prove useful. This is what we now propose in what follows.

Looking into the potential / limitations of tandem learning initiatives to foster professionalization: analyzing data from the field

In order to better understand the pedagogical and professional benefits / weaknesses of tandem learning to foster skill development, three tandem / collaborative pre-service language teacher training projects are described and analyzed.

Presentation of the projects

Table 4 below is an outline of the main features which characterize the three joint Table 4. A synthetic outlook of the 3 Franco-German tandem learning / group work projects.

The Siegen-Paris project

The aim of the "Siegen-Paris" project (Abendroth-Timmer and Aguilar Río 2014) was to have a group of pre-service language teachers (n = 21) develop their digital literacy skills (Abendroth-Timmer and Aguilar, 2014). To do so, 21 master's trainee teachers enrolled at Sorbonne Nouvelle (n = 11) and Siegen (n =10) formed 9 tandems and one tridem during a 12-week term. The participants were led to collaboratively produce 5 written summaries of 5 articles chosen from a suggested corpus of scientific journal articles (in English, French, German and Spanish) in applied linguistics and multiliteracy. In this project, it was expected that a collaborative work environment would help the participants stand back from their (un)conscious beliefs about language learning / teaching and the use of technology and that, in so doing, they would be encouraged to both enhance their learning processes and develop their teaching practices. Apart from having trainee teachers discover and manipulate various technological tools that can be used for pedagogical purposes, it was expected that the implemented tandem dynamics would help the technically challenged participants of the cohort overcome -or, at least, nuance -their potential concerns regarding the use of digital tools. Pre-and post-experience surveys were administered to identify indicators of change among the participants as regards their beliefs about digital literacy and language learning / teaching.

The Berlin-Paris-Siegen project

The "Berlin-Paris-Siegen" project included -on top of the two above-mentioned institutional partners -the Humboldt University (Berlin) and two secondary schools:

one located in La Rochelle (France) and one in Berlin. 20 master's trainee teachers, enrolled at Sorbonne Nouvelle (n = 10), Siegen (n = 5) and the Humboldt (n = 5) universities were grouped into 10 tandems. The pursued learning outcome of this project was to have the study participants develop their professional awareness about the intercultural dimension underlying any language learning / teaching situation [START_REF] Kramsch | Intercultural Contact, Hibridity and Third Space[END_REF]. For this project, new elements were added to the core of the "Siegen-Paris" project: after completing a series of task which instructions were similar to that of the Siegen-Paris project over a 12 week period, the study participants had to rely upon the pedagogical and language learning related ideas, principles and theories brought forward in the four summaries they had produced to design -during an additional second 13-week term -learning scenarios in French or German which had to focus on the cultural features of one of these two languages 7 . Depending on the target language of the tasks, these were then submitted to the learners of French in Berlin or the learners of German in La Rochelle. The learners were, in turn, required to work under the guidance of their teacher to complete the scenarios which had been prepared

for them by the pre-service teachers (who also happened to serve as online tutors). In this project, it was expected that collaborative work would lead the trainee teachers to take some distance from their (un)conscious beliefs about the "cultural weight"

underlying any given learning and teaching language situation and, more importantly, to benefit from a first-hand professional experience of online language teaching which they could potentially replicate or use as a working basis later on during their teaching careers. Again, the participants were asked to complete pre-and post-experience surveys, to try and identify indicators of change concerning their beliefs and skills.

7 For instance, one of the proposed tasks focused on how to express opinions and (dis)likes in French and to reflect upon one's cultural identity, by discussing two posters corresponding to two French films, as well as reacting to several excerpts taken from these films.

The CONFORME project

The third project, CONFORME 8 (Brudermann et al. in press;[START_REF] Xue | Interaction En Formation Des Enseignants de Langue : Sensibilisation de Futurs Enseignants Sur Le Plan Émotionnel Par Un Dispositif Hybride Interactif[END_REF], gathered 54 undergraduate and, for most of them, master's trainee teachers enrolled at Sorbonne Nouvelle (n = 43) and Siegen Universities (n = 11). As a consequence of the unequal number of participants at each site, a collaborative work modality was chosen as the most adequate option to consider to reach the objectives which were set, i.e. to develop the study participants' professional awareness of the emotional aspects underlying any language learning and teaching situation (MacIntyre, Gregersen, and Mercer 2016). The participants teamed up in six 4-member groups and six 5-member groups. During a 12-week term, they were led to collaboratively complete three collective tasks -i.e. to produce summaries of two scientific articles and to devise a language teaching observation grid -and submit one final, individual task, i.e. a reflexive account of their experience during the course of the module. During a second 13-week term, it was expected that the participants would rely upon their observation grids to analyse their own as well as their peers' teaching practices, as a selfconfrontation and reconstruction method of sorts [START_REF] Tanghe | Build[] Something Which Alone We Could Not Have Done': International Collaborative Teaching and Learning in Language Teacher Education[END_REF]. As it was the case with the two previous projects, it was expected that the collective work dynamics would fulfil a mediating function that would lead the participants to take some distance from their (un)conscious beliefs regarding language learning and teaching. Besides, the situated, goal-oriented verbalization protocol which had been prepared for the second part of the project was intended as a further mediating tool, insofar so as its aim was to have each participant experience a constructive, critical and holistic foreign language teaching situation. The participants were eventually required to complete pre-and postexperience surveys, to try and identify indicators of change (Brudermann et al. in press) 8 A synthetic, French description of the project can be retrieved at http://tinyurl.com/htnrjx2

(last accessed on November, 23th 2017).

in their professional practices / beliefs.

Perceived commitment of the participants and attitude regarding the projects

One common feature to the three projects was the implementation of a blended environment [START_REF] Gascón | Aprendizaje cooperativo y evaluación formativa en marketing: tándem ganador en docencia universitaria[END_REF] featuring tools that allowed for both synchronous and asynchronous types of communication to take place. The participants' answers to the post-experience surveys which were carried out in the three projects (Abendroth-Timmer and Aguilar Río 2014; [START_REF] Xue | Interaction En Formation Des Enseignants de Langue : Sensibilisation de Futurs Enseignants Sur Le Plan Émotionnel Par Un Dispositif Hybride Interactif[END_REF]Brudermann et al. in press) indicated a consensus among the participants which revolved around the perceived relevance of asynchronous modes of communication to reach the objectives which were set insofar as these tools allow to alleviate the communicative pressure that can potentially be felt during real-time, face-to-face exchanges 9 . Another form of consensus emerging from the participants' responses in the three studies has to do with their overall positive impressions regarding the blended environments proposed: these apparently provided them with enough leeway to proceed to self-regulated learning strategies and seemed to suit their needs, as members of given communities of practice [START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF].

Specific difficulties to either or all of the three projects were also met by both the participants and the research teams (Abendroth-Timmer and Aguilar Río 2014).

These coincide with the three main categories that emerged from the analysis of the subcorpus which we proposed above, namely: (i) relationship issues among the participants, (ii) a need for adaptable work environments and (iii) the methodological choices and their relation to identifiable indicators of change among the participants. 9 Even though real time exchanges were at the same time considered important to trigger social reliance [START_REF] Aden | Teaching Languageand Culture in an Era of Complexity: Interdisciplinary Approaches for an Interrelated World / Enseigner les langues-cultures à l'ère de la complexité: Approches interdisciplinaires pour un monde en reliance[END_REF].

Relationship issues among the participants

In all three cases, failure to comply with [START_REF] Gascón | Aprendizaje cooperativo y evaluación formativa en marketing: tándem ganador en docencia universitaria[END_REF] suggestions occurred, that is to guide the members of a tandem or group in such a way that they can carry out an introspective work that allows them to identify individual capabilities, strengths and talents, which they can, in turn, rely upon to serve the common, collective interests and objectives of the group. For the three projects, the physical encounters took place after the pre-experience surveys had been filled out by the participants. The research members in charge of the projects used the participants' answers to the pre-experience surveys to identify information that suggested potential affinities among particular members and further form the groups. This method proved adequate for the "Siegen-Paris" and "Berlin-Paris-Siegen" projects, in which the ratio of participants per tutor was respectively 21:2 and 20:3. During the CONFORME project, the 54:3 ratio placed high workload demands on the tutors, particularly as regards their capacity to follow up on the participants' progressions in the tasks they were assigned.

Such an unbalanced ratio may have led to the participants' deployment of coping strategies, which could have, in turn, fostered the development of specific skills, such as self-directed learning and making one's initiatives known and valuable within the group 10 . Finally, the way the groups were formed also took into consideration the sociocultural backgrounds and L1(s) of the participants. The aim was to implicitly give French a lingua franca status within the groups of students 11 , who came from Arabic, Chinese, French, German and Hispanic speaking and educational backgrounds, even 10 Ramona Schneider, a CONFORME research team member, is currently drafting a PhD dissertation that should confirm this hypothesis.

11 As most of the students in the project were enrolled in a French institution and as the rest of the participants in Germany were specialised in romance languages, is was estimated that leading the participants to use French as a vehicular language was the most relevant option to consider in this project to have them work collaboratively.

though the participants were free to draw on their linguistic repertoires to carry out their assignments. Another decisive criterion which may have been worth taking into account to form the groups -and which was later suggested by the participants themselveswas their prior training and previous teaching experiences. In effect, occurrences of phenomena such as unequal commitment (Guichon 2009, 180), declared incompatibilities between specific members within a group, or conflicts which were made explicit and had to be dealt with, were reported several times during the projects.

Should a certain period of time have been allotted -prior to the actual work -for the participants to make their own choices, such relationship issues might well have been prevented.

A need for adaptable work environments

As a consequence of the previous perceived difficulties, the work environments were designed and implemented so as to encourage the participants to take initiative. By "work environment", we mean, not only the technical tools, solutions and implementations that were made accessible to the participants at a distance to come together and collaborate, but also the institutional frame sustaining such collaborative work -namely the chronological organisation of the terms, the assessment criteria and the different grading systems at each site -as well as the pedagogical contents 12 . This "openness" turned out to be a necessary condition for the "Berlin-Siegen-Paris" and CONFORME projects: because of the expected learning outcomes of these two projects, the research team deemed it necessary to run them over two terms, to endow each experience with sufficient longitudinal scope so as to further be able to better identify indicators of change among the participants. In the case of the "Berlin-Paris-Siegen" project, the adaptability of the work environment proved robust when the fourth external partner -a teacher of German in a secondary school in La Rochelle -dropped out. In this particular case, arrangements were made for the trainee teachers to nonetheless stay in contact with the learners in Berlin, so the latter could at least receive and try to complete the tasks that the former had developed for them.

In the case of the CONFORME project, a major obstacle was met when 40 participants (out of the total 43 who were enrolled in Paris) dropped out at the end of the first term. Their decision was justified by the fact that they no longer had an institutional obligation to maintain their collaboration at that stage, which henceforth had to continue on a voluntary basis. Although most participants regarded the project positively, as far as their prospective careers as language teachers were concerned (Brudermann et al. in press), the perceived institutional workload that awaited them in other modules during the second term of their master's -and which was to be assessed and graded -fatally jeopardised their involvement in the second half of the project.

Thus, the adaptability of the CONFORME project proved successful when new groups were formed among the 14 remaining participants in Paris (N = 3) and Siegen (n =11).

The technical tools and solutions to which the participants had grown accustomed during the first term allowed them to team up with new partners, efficiently collaborate to perform the assigned tasks and to amend some of those they had completed during the first term with other partners to adjust them to the objectives of the second phase of the project.

Methodological choices

To a certain extent, each of the three discussed projects was an upgraded version of the preceding one. As such, elements of continuity among the three projects can be noted, as is the case for example for some of the research team members. The outlook of the projects also shared some elements of continuity, as their scopes included multicultural, multilingual and multimedia dimensions. Eventually, as far as the research methodology employed, the participants were required to complete pre-and post-experience surveys.

However, since each project aimed at different learning outcomes (cf. table 4), specific work procedures were implemented. For the "Berlin-Paris-Siegen" project, the instructions for the participants were drafted so as to drive them into leaving traces of their (a)synchronous exchanges, which could henceforth be used for research purposes 13 . As for the CONFORME project, some specific technical solutions were implemented to allow the participants not only to carry out a posteriori verbalizations, but also to keep traces of these self-confrontation and reconstruction experiences. As stated above, these technological choices met a major institutional obstacle when most participants in Paris decided to drop out at the end of the first phase of the project.

Offering built-in online environments concentrating all sorts of potentially useful digital tools to carry out the assignments in one single online location seemed less important for the users: each group indeed used the offered environments differently and sometimes freely resorted to "external" tools 14 , such as Skype, their e-mails, Google Drive, Facebook, or Whatsapp. This apparently depended on individual preferences concerning variables such as (a)synchronous communication, written / oral communication or individual time availability and division of work.

Indicators of professional development

If we refer to the three tandem learning experiences we have introduced, evidence from the field -and, in particular, the self-confrontation and reconstruction reports which the participants were required to draft during the second phase of the CONFORME project -seems to reveal that evident gains can be outlined, as far as skill development is concerned. Indeed, it appears that the participants:

 Learnt to choose, use and individually adapt a diversity of digital tools, notably for communication purposes. This process led them to reflect upon their individual preferences in terms of ICT use and to enhance both their competence and knowledge in the field of multimedia;

 Got accustomed to collaborating within multicultural and multilingual (online)

environments. This, in particular, drove them into learning how to solve relationship problems and, in so doing, develop their socio-cultural and emotional skills;

 Were confronted with culturally-influenced learning / working traditions and communication strategies. The projects therefore also had them not only experience but also question the academic traditions they were in touch with and, in so doing, enhance their methodological skills;

 Were equipped with and trained to use a set of robust scientific tools (supposedly) able to have them develop their cognitive competence, gain knowledge in the field of language education and inform their practice. This helped them reflect on the theoretical input which was provided in a broader international education macrocosm and widen their theoretical backgrounds.

They can now put these skills to good use at a later stage to refine their analyses of language education situations;

 Were taught how to use and reinvest first-hand methodological tools able to promote professional self-reflexion and facilitate a call into question of their skills, beliefs and attitudes towards language learning and teaching. Besides, most tasks in the project led them to establish strong links between what was at stake in the course and their former learning and teaching experiences. This approach fostered reflection and professional competence development (notably for lifelong learning) and acted as a lever to have them both develop a new theoretical understanding and be endowed with new practical teaching perspectives.

Discussion

Both tandem and group work projects share the fundamental feature of bringing within an educational context a particular social dimension, in order to promote skill / knowledge development in a specific subject (cf. Table 2). Tandem initiatives often take the form of learning partnerships between two people with different languages, which some authors refer to as dyads [START_REF] Guichon | Training Future Language Teachers to Develop Online Tutors' Competence through Reflective Analysis[END_REF][START_REF] Tanghe | Build[] Something Which Alone We Could Not Have Done': International Collaborative Teaching and Learning in Language Teacher Education[END_REF]. The difference between tandems and dyads seems to lie in the status -be it perceived or attributed -of the members working together within a group. While tandems seem to point towards a form of equality, "dyads" suggest a differential hierarchy between the members, powered by their differing degrees of expertise in relation with a skill or learning objective. There also seems to be more room within the tandem dynamics for the participants to take initiatives, use their creativity and share their enthusiasm. These three features appear to be associated with some forms of autonomy and self-regulation which seem to be compatible with the tandem dynamics. However, in tandem initiatives, the duration of the projects may be extremely variable and, in some cases, this parameter may endanger the very continuation of the initiatives themselves. A such, in order to circumvent this difficulty, the intervention of an institutional third partysuch as a tutor -who may help (re)define the learning objectives of the work groups, as well as the workings of their dynamics -is required in tandem learning projects. In particular, group work entails an association made up of more than two members.

Although unequal perceived commitment is not unknown to tandems, the feedback from experience gained in the three projects we have conducted seems to reveal that group work requires an explicit negotiation of the members' perceived skills and strengths before the onset of the projects, to further put them at the service of all the participants. Such a negotiation -which is meant to allow the necessary introspection and communication processes to take place -can be achieved through the implementation of (intercultural) mediation, endorsed by third party tutors.

To take this variable into account, the tandem and group work initiatives we designed sought to propose adaptable work environments and conditions, for example by allowing for adjustments on the go to be made. At the same time, the management of such of pedagogical practice came with a significant workload, both for the participants and for administrators -be they involved as tutors or researchers. Accordingly, technical and organisational choices were made to make the work environment as welcoming and user-friendly as possible, but also to better handle the additional workload. In particular, evidence from the field shows that the goodwill of the course managers seems capital to provide a (viable) learning structure, give positive feedback and motivation and help resolving technical, cognitive or even social problems.

Eventually, when implementing the tandem or group work environments presented above, we tried to do so by taking into account the learners' specificities or choosing an instructional stance allowing to assess the potential of the devised working environments to reach the objectives they sought to pursue (Brudermann et al. in press).

This is why, in our review, special attention was paid to (i) the pedagogical engineering aspects underlying the implementation of our tandem learning / group work initiatives, (ii) the tools and methodological choices we made to assess the relevance of the work environments which were designed to foster professional skill development and (iii) the educational potential of the implementation of a multicultural and multilingual collaborative work environment to train pre-service language teachers. What emerged from the three projects presented is that these various pedagogical initiatives seemamong other things -to have strong links with Vygotsky's (still) pervasive sociocultural theory (1978). In particular, what appears to be a common element is the principle that learning forms a socially contextualised framework and that this effort is liable to foster both the emergence of new knowledge 15 and skill development.

Thus, as far as language teacher training is concerned, our findings suggest that replacing teacher-fronted pedagogical practices by collaborative practices that allow for the participants to construct their knowledge may further facilitate the training (i.e. the acquisition / development of professional skills) as well as engage the participants more sustainably.

Tandem learning: pedagogical innovation as a recycling endeavour

In this paper, an exploration -both from a theoretical and a practical point of view -of the potential and limitations of tandem learning to foster professional development amongst pre-service language (L2) teachers was proposed. The research works reviewed in this study underscore their adequacy as frameworks that allow the pursuit of specific learning outcomes in genuine social contexts, like, for instance, change within an individual or the formation of communities of practice that federate groups of individuals who temporarily share common goals. Another common feature which seems to stand out as a strong link between the studies we have reviewed and our own projects is "collaboration", which is coherent with the Vygotskian perspective we have just dealt with, since the central goal of collaborative learning is the common construction of knowledge / understanding [START_REF] Littleton | Learning Together: Understanding the Processes of Computer-Based Collaborative Learning[END_REF].

Specific relational, technological and methodological difficulties have also been identified. They seem to appear as features which are particular to the complex dynamics that both tandem learning and group work calls into play. The discussion that followed the presentation of the projects we have carried out has allowed for these difficulties to be contextualised and illustrated and examples of the ad hoc solutions which were proposed to take these difficulties into account were also provided. For example, we estimated that the technological choices underlying the implementation of a tandem learning project should only come into play once a sufficient diagnosis of the participants' (perceived) needs, strengths and expectations has been performed.

Eventually -and above all -resistance in the implementation of tandem learning initiatives may come from the institution, namely some of its stakeholders and policy makers. Tandem and group work experiences entailing forms of international collaboration indeed require to make do with different administration management systems, pedagogical priorities -and, by subsequent effect, assessment criteria -and to prepare legal documents in various languages (sometimes pertaining to different legal traditions). The bureaucratic-pedagogical background that underlies any tandem learning initiative therefore often lays down heavy constraints for the pedagogical staff in charge of such projects. These may sometimes jeopardize the very implementation of the projects -given the very important amount of extra (and most of the time unpaid) work they imply for the staff members -and, in turn, prove paradoxically counterproductive for the institutions themselves, as such pedagogical "nonhappenings" are liable to impair the institutions' international recognition. In any case, experience from the field seems to show that, besides these obstacles, in order to promote the smooth functioning of a tandem learning initiative, setting clear agendas, providing guidance and ensuring transparency towards all the stakeholders are essential steps to follow. As such, working with different stakeholders from various educational backgrounds may prove constraining. But more potential benefits can be withdrawn by working with them, rather than in spite of them.

  [START_REF] Tanghe | Build[] Something Which Alone We Could Not Have Done': International Collaborative Teaching and Learning in Language Teacher Education[END_REF], who underline the importance for the participants to feel familiar with the technology(ies) employed in the learning environment to smooth out the collective work processes underlying any tandem learning / group work effort. The tandem / group work set up may also be a source of discomfort for the participants. As a consequence, the authors stress the importance of facilitating contact among participants prior to specific tandem work to make things work (as far as possible).As concerns the management of a tandem learning experiment, Gascón and De-Juan-Vigaray (2015, 376) insist on the energy and time investment required from the tutoring team. These authors insist on the importance for the tutors and supervisors to keep an open mind and make the potential necessary changes to the working environment as the learning experience unfolds.Xue and Schneider (2015, 14) also highlight the importance for the research team to look at the work environment critically, since they found that what the participants said they had experienced led to the reinforcement of some of their prior (un)conscious beliefs.

  language teacher training and research projects which we will focus on in what follows and which brought together several French and German learning institutions: Sorbonne Nouvelle and Siegen universities, the Humboldt University of Berlin and two secondary schools (one in La Rochelle (France) and the other in Berlin (Germany)). For these projects, both tandem learning and group work modalities were implemented. for "Cognitions, émotions et médiations en formation des enseignants de langues".

12

  It was estimated by the pedagogical teams of the projects that the work instructions hat to be clear, precise and open, so as to encourage the participants to use their creativity, initiative and previous knowledge during task completion.

Table 3 .

 3 References explicitly referring to tandem learning in the corpus.

	5	A corresponding Zotero collection is available online (cf. http://tinyurl.com/y9vb8akm,
		last accessed 24th October 2017).

All participants in each of the three projects were required to read and sign a document, either to accept or to refuse the conditions that stated how the data produced would be processed, anonymised, analysed and ultimately made public for research-related and teacher training purposes.


Feng Qi

AND Ai-Qi Liu

Dongkyu Lim
email: dklim@andong.ac.kr

EXPLICIT EXPRESSIONS RELATED TO DEGENERATE CAUCHY NUMBERS AND THEIR GENERATING FUNCTION

Keywords: 2010 Mathematics Subject Classification. Primary 11B68, Secondary 11B83, 33B10, 33B99, 34A05, 34A34 explicit expression, degenerate Cauchy number, coefficient, nonlinear differential equation, generating function, Bell polynomial of the second kind, Faà di Bruno formula

In the paper, by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of the second kind, the authors establish an explicit expression for degenerate Cauchy numbers and find explicit, meaningful, and significant expressions for coefficients in a family of nonlinear differential equations for the generating function of degenerate Cauchy numbers.

Motivations and main results

It is well known [START_REF] Qi | A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind[END_REF][START_REF] Qi | An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind[END_REF][START_REF] Qi | Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind[END_REF][START_REF] Qi | Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers[END_REF][START_REF] Qi | An integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind[END_REF] that the Cauchy numbers of the first kind C n can be generated by

t ln(1 + t) = ∞ n=0 C n t n n! .
In [START_REF] Kim | Differential equations associated with degenerate Cauchy numbers[END_REF], degenerate Cauchy numbers C n (λ) were defined by λ e [(1+t) λ -1]/λ -1

(1 + t) λ -1 = ∞ n=0
C n (λ) t n n! .

Because lim

(1 + t) λ -1 λ = ln(1 + t) or lim λ→0 e [(1+t) λ -1]/λ = 1 + t, it follows that lim

λ→0 C n (λ) = C n , n ≥ 0. (1)
In [2, Theorem 2.1], it was established that the family of nonlinear differential equations

(1+t) n (1+t) λ -1 n F (n) λ (t) = F λ (t) 2n i=1 a i (n, λ)(1+t) iλ + 2n-1 i=1 b i (n, λ)(1+t) iλ (2)
for n ∈ N has the same solution

F λ (t) = e [(1+t) λ -1]/λ -1 (1 + t) λ -1 , (3)
where a i (n, λ) for 1 ≤ i ≤ 2n and b i (n, λ) for 1 ≤ i ≤ 2n-1 are uniquely determined by

a 1 (n, λ) = - 1 λ n -1 -λ n+1 , a 2 (n, λ) = n -1 -2λ n-1 - 1 λ n-2 i=0 [λ -(λ + 1)(n -i)] n -i -2 -λ n-i n -1 -2λ i , a i (n, λ) = [(i -1)λ -(λ + 1)n]a i (n -1, λ) +a i-2 (n -1, λ) + (n -1 -iλ)a i (n -1, λ), 3 ≤ i ≤ 2n -2, a 2n-1 (n, λ) = 1 2 n[(λ -1)(n -1) -2(λ + 1)], a 2n (n, λ) = 1, b 1 (n, λ) = n -1 -λ n-1 , b i (n, λ) = [(i -1)λ -(λ + 1)(n -1)]b i-1 (n -1, λ) +a i-1 (n -1, λ) + (n -1 -iλ)b i (n -1, λ), 2 ≤ i ≤ 2n -3, b 2n-2 (n, λ) = (λ -1) n -1 2 -2(n -1) -λ, b 2n-1 (n, λ) = 1
in terms of the falling factorials

x n = n-1 k=0 (x -k) = x(x -1) • • • (x -n + 1), n ≥ 1; 1, n = 0.
It is clear that the generating function F λ (t) defined by (3) satisfies

lim λ→0 [λF λ (t)] = t ln(1 + t) .
It is obvious that

(1) the above expressions for a i (n, λ) and b i (n, λ) are recursive and can not be computed easily; (2) the original proof of [2, Theorem 2.1] is inductive, recursive, and long;

(3) there was no any application given in [START_REF] Kim | Differential equations associated with degenerate Cauchy numbers[END_REF].

In this paper, by virtue of the Faà di Bruno formula [START_REF] Qi | Explicit formulas for the convolved Fibonacci numbers[END_REF] and two identities [START_REF] Qi | Notes on several families of differential equations related to the generating function for the Bernoulli numbers of the second kind[END_REF] and [START_REF] Qi | Simple forms for coefficients in two families of ordinary differential equations[END_REF] for the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n-k+1 ), we will establish an explicit expression (4) for degenerate Cauchy numbers C n (λ) and find explicit, meaningful, and significant expressions [START_REF] Qi | An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind[END_REF] and [START_REF] Qi | Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind[END_REF] for coefficients a i (n, λ) and b i (n, λ) in the family of nonlinear differential equations (2) related to the generating function F λ (t) of degenerate Cauchy numbers C n (λ).

Our main results can be stated as the following theorems.

Theorem 1. For n ≥ 0, degenerate Cauchy numbers C n (λ) and the Cauchy numbers C n can be explicitly and respectively computed by

C n (λ) = n k=0 (-1) k (k + 1)!λ k k =0 (-1) k λ n (4)
and

C n = n k=0 s(n, k) k + 1 . ( 5
)
Theorem 2. For n ∈ N, the generating function F λ (t) and its derivatives of degenerate Cauchy numbers C n (λ) satisfy

F (n) λ (t) = F λ (t) (1 + t) n 1 -(1 + t) λ n 2n i=1 α i (n, λ)(1 + t) iλ + 2n-1 i=1 β i (n, λ)(1 + t) iλ (6) with α i (n, λ) = k+m=i 1≤k≤n 0≤m≤n (-1) m A k (n, λ) min{n-m,k} =0 λ (k -)! n - m (7)
for 1 ≤ i ≤ 2n and

β i (n, λ) = k+m=i 1≤k≤n 0≤m≤n-1 (-1) m+1 A k (n, λ) min{k-1,n-m-1} =0 λ (k -)! n --1 m (8)
for

1 ≤ i ≤ 2n -1, where A k (n, λ) = (-1) k λ k k =0 (-1) k n-1 q=0 ( λ -q).

Lemmas

In order to obtain our main results, we need the following lemmas.

Lemma 1 ([1, pp. 134 and 139]). For n ≥ k ≥ 0, the Bell polynomials of the second kind, denoted by B n,k (x 1 , x 2 , . . . , x n-k+1 ), are defined by

B n,k (x 1 , x 2 , . . . , x n-k+1 ) = 1≤i≤n-k+1 i∈{0}∪N n-k+1 i=1 i i=n n-k+1 i=1 i=k n! n-k+1 i=1 i ! n-k+1 i=1 x i i! i .
The Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n-k+1 ) by

d n d t n f • h(t) = n k=0 f (k) (h(t)) B n,k h (t), h (t), . . . , h (n-k+1) (t) . (9)
Lemma 2 ([1, p. 135]). For n ≥ k ≥ 0, we have

B n,k abx 1 , ab 2 x 2 , . . . , ab n-k+1 x n-k+1 = a k b n B n,k (x 1 , x 2 , . . . , x n-k+1 ), ( 10
)
where a and b are any complex numbers.

Lemma 3. For n ≥ k ≥ 0 and λ, α ∈ C, we have

B n,k 1, 1 -λ, (1 -λ)(1 -2λ), . . . , n-k =0 (1 -λ) = (-1) k k! k =0 (-1) k n-1 q=0 ( -qλ) (11) or, equivalently, B n,k ( α 1 , α 2 , . . . , α n-k+1 ) = (-1) k k! k =0 (-1) k α n . (12)
Proof. This explicit formula [START_REF] Qi | Simple forms for coefficients in two families of ordinary differential equations[END_REF]

Proofs of Theorems 1 and 2

We are now in a position to prove our main results.

Proof of Theorem 1.

For n ≥ 0, applying u = h(t) = (1+t) λ -1 λ and f (u) = e u -1
u to ( 9) and making use of ( 10) and [START_REF] Qi | Simple forms for coefficients in two families of ordinary differential equations[END_REF] in sequence arrive at

d n [λF λ (t)] d t n = n k=0 d k d u k e u -1 u B n,k λ(1 + t) λ-1 λ , λ(λ -1)(1 + t) λ-2 λ , . . . , λ(λ -1) • • • [λ -(n -k)](1 + t) λ-(n-k+1) λ = n k=0 d k d u k ∞ =1 u -1 ! (1 + t) kλ-n B n,k (1, λ -1, . . . , (λ -1) • • • [λ -(n -k)]) = n k=0 d k d u k ∞ =0 u ( + 1)! (1 + t) kλ-n × B n,k 1, λ 1 - 1 λ , . . . , λ n-k 1 - 1 λ • • • 1 - n -k λ = n k=0 ∞ =0 d k d u k u ( + 1)! (1 + t) kλ-n × λ n-k B n,k 1, 1 - 1 λ , . . . , 1 - 1 λ • • • 1 - n -k λ = n k=0 ∞ =k ! ( -k)! u -k ( + 1)! (1 + t) kλ-n λ n-k (-1) k k! k =0 (-1) k n-1 q=0 - q λ = n k=0 ∞ =k 1 ( -k)! u -k + 1 (1 + t) kλ-n 1 λ k (-1) k k! k =0 (-1) k n-1 q=0 ( λ -q) → n k=0 1 k + 1 1 λ k (-1) k k! k =0 (-1) k n-1 q=0 ( λ -q) = n k=0 (-1) k (k + 1)!λ k k =0 (-1) k λ n
as t → 0 and, consequently, u → 0. This implies that

C n (λ) = λ lim t→0 F (n) λ (t) = n k=0 (-1) k (k + 1)!λ k k =0 (-1) k λ n .
The explicit formula ( 4) is thus proved. It is well known [1, Theorem A] that the Stirling numbers of the first kind s(n, k) can be generated by

x n = n k=0 s(n, k)x k .
Hence, by the L'Hôspital rule, we have

lim λ→0 1 λ k k =0 (-1) k λ n = lim λ→0 1 λ k k =0 (-1) k n m=0 s(n, m)( λ) m = 1 k! lim λ→0 k =0 (-1) k n m=0 s(n, m) m k k ( λ) m-k = 1 k! k =0 (-1) k s(n, k) k k k = s(n, k) k =0 (-1) k k = (-1) k k!s(n, k).
Combining this with ( 1) and ( 4) gives

C n = lim λ→0 C n (λ) = n k=0 (-1) k (k + 1)! lim λ→0 1 λ k k =0 (-1) k λ n = n k=0 s(n, k) k + 1 .
The explicit formula (5) follows. The proof of Theorem 1 is complete.

Proof of Theorem 2. For n ∈ N, applying u = h(t) = (1+t) λ -1 λ and f (u) = e u -1 u to ( 9) and making use of ( 10) and [START_REF] Qi | Simple forms for coefficients in two families of ordinary differential equations[END_REF] in sequence arrive at

d n [λF λ (t)] d t n = n k=1 d k d u k e u -1 u B n,k λ(1 + t) λ-1 λ , λ(λ -1)(1 + t) λ-2 λ , . . . , λ(λ -1) • • • [λ -(n -k)](1 + t) λ-(n-k+1) λ = n k=1 k =0 k (e u -1) ( ) 1 u (k-) (1 + t) kλ-n × B n,k (1, λ -1, . . . , (λ -1) • • • [λ -(n -k)]) = n k=1 (-1) k k!(e u -1) u k+1 + k =1 k e u (-1) k-(k -)! u k-+1 (1 + t) kλ-n × B n,k 1, λ 1 - 1 λ , . . . , λ n-k 1 - 1 λ • • • 1 - n -k λ = n k=1 λF λ (t) k =0 k (-1) k-(k -)! u k- + k =1 k (-1) k-(k -)! u k-+1 (1 + t) kλ-n × λ n-k B n,k 1, 1 - 1 λ , . . . , 1 - 1 λ • • • 1 - n -k λ = n k=1 λF λ (t) k =0 k (-1) k-λ k-(k -)! [(1 + t) λ -1] k- + k =1 k (-1) k-λ k-+1 (k -)! [(1 + t) λ -1] k-+1 × (1 + t) kλ-n λ n-k (-1) k k! k =0 (-1) k n-1 q=0 - q λ = 1 (1 + t) n n k=1 λF λ (t) k =0 k (-1) λ ! [(1 + t) λ -1] + k-1 =0 k (-1) λ +1 ! [(1 + t) λ -1] +1 × (1 + t) kλ λ k (-1) k k! k =0 (-1) k n-1 q=0 ( λ -q).
Accordingly, we have

(1 + t) n (1 + t) λ -1 n F (n) λ (t) = n k=1 (1 + t) kλ λ k (-1) k k! k =0 ( -1) k n-1 q=0 ( λ -q)
× F λ (t)

k =0 k (-1) λ ![(1 + t) λ -1] n-+ k-1 =0 k (-1) λ ![(1 + t) λ -1] n--1 = n k=1 (-1) k λ k k =0 (-1) k n-1 q=0 ( λ -q) (1 + t) kλ × F λ (t) k =0 (-1) λ (k -)! n- m=0 n - m (1 + t) mλ (-1) n--m + k-1 =0 (-1) λ (k -)! n--1 m=0 n --1 m (1 + t) mλ (-1) n--1-m = (-1) n n k=1 (-1) k λ k k =0 (-1) k n-1 q=0 ( λ -q) (1 + t) kλ × F λ (t) k =0 λ (k -)! n- m=0 (-1) m n - m (1 + t) mλ + k-1 =0 λ (k -)! n--1 m=0 (-1) m+1 n --1 m (1 + t) mλ = (-1) n F λ (t) n k=1 A k (n, λ) k =0 λ (k -)! n- m=0 (-1) m n - m (1 + t) (k+m)λ + (-1) n n k=1 A k (n, λ) k-1 =0 λ (k -)! n--1 m=0 (-1) m+1 n --1 m (1 + t) (k+m)λ . Since n k=1 A k (n, λ) k =0 λ (k -)! n- m=0 (-1) m n - m (1 + t) (k+m)λ = n k=1 A k (n, λ) k =0 λ (k -)! n m=0 (-1) m n - m (1 + t) (k+m)λ = n k=1 A k (n, λ)(1 + t) kλ n m=0 (-1) m k =0 λ (k -)! n - m (1 + t) mλ = n k=1 A k (n, λ)(1 + t) kλ n m=0 (-1) m min{n-m,k} =0 λ (k -)! n - m (1 + t) mλ = n k=1 A k (n, λ) n m=0 (-1) m min{n-m,k} =0 λ (k -)! n - m (1 + t) (k+m)λ = 2n i=1       k+m=i 1≤k≤n 0≤m≤n (-1) m A k (n, λ) min{n-m,k} =0 λ (k -)! n - m       (1 + t) iλ and n k=1 A k (n, λ) k-1 =0 λ (k -)! n--1 m=0 (-1) m+1 n --1 m (1 + t) (k+m)λ = n k=1 A k (n, λ) k-1 =0 λ (k -)! n-1 m=0 (-1) m+1 n --1 m (1 + t) (k+m)λ = n k=1 A k (n, λ) n-1 m=0 (-1) m+1 k-1 =0 λ (k -)! n --1 m (1 + t) (k+m)λ = n k=1 A k (n, λ) n-1 m=0 (-1) m+1 min{k-1,n-m-1} =0 λ (k -)! n --1 m (1 + t) (k+m)λ = 2n-1 i=1       k+m=i 1≤k≤n 0≤m≤n-1 (-1) m+1 A k (n, λ) min{k-1,n-m-1} =0 λ (k -)! n --1 m       (1 + t) iλ ,
where an empty sum is understood to be 0 and p q = 0 for q > p ≥ 0, the equation ( 6) and the formulas (7) and 8 are thus proved. The proof of Theorem 2 is complete.

Remarks

Finally, we list several remarks on our main results and closely related things. Remark 1. Comparing ( 6) with (2) reveals that a i (n, λ) = (-1) n α i (n, λ) and b i (n, λ) = (-1) n β i (n, λ).

Remark 2. It is easy to see that explicit expressions [START_REF] Qi | An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind[END_REF] and ( 8) for α i (n, λ) and β i (n, λ) are more meaningful and more significant than those in [2, Theorem 2.1] for a i (n, λ) and b i (n, λ) mentioned above.

Remark 3. The formula (5) was derived in [START_REF] Nemes | An asymptotic expansion for the Bernoulli numbers of the second kind[END_REF] and mentioned in [START_REF] Qi | A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind[END_REF][START_REF] Qi | Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind[END_REF].

Remark 4. Per requests of anonymous referees, the preprint [START_REF] Qi | On tridiagonal determinants and the Cauchy product of central Delannoy numbers[END_REF] is split into and simplified as two formally published papers [START_REF] Qi | Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials[END_REF][START_REF] Qi | Alternative proofs of some formulas for two tridiagonal determinants[END_REF].

Remark 5. The motivations in the papers [START_REF] Qi | A simple form for coefficients in a family of nonlinear ordinary differential equations[END_REF][START_REF] Qi | A simple form for coefficients in a family of ordinary differential equations related to the generating function of the Legendre polynomials[END_REF][START_REF] Qi | Explicit formulas for the convolved Fibonacci numbers[END_REF][START_REF] Qi | Notes on several families of differential equations related to the generating function for the Bernoulli numbers of the second kind[END_REF][START_REF] Qi | Simple forms for coefficients in two families of ordinary differential equations[END_REF][START_REF] Qi | Simplification of coefficients in two families of nonlinear ordinary differential equations[END_REF][START_REF] Qi | Simplifying coefficients in a family of nonlinear ordinary differential equations[END_REF][START_REF] Qi | Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials[END_REF][START_REF] Qi | Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Mittag-Leffler polynomials[END_REF][START_REF] Qi | Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials[END_REF][START_REF] Qi | A diagonal recurrence relation for the Stirling numbers of the first kind[END_REF][START_REF] Qi | Explicit formulas and recurrence relations for higher order Eulerian polynomials[END_REF][START_REF] Qi | Some properties of the Hermite polynomials[END_REF][START_REF] Qi | Viewing some ordinary differential equations from the angle of derivative polynomials[END_REF][START_REF] Qi | Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations[END_REF][START_REF] Qi | Some identities related to Eulerian polynomials and involving the Stirling numbers[END_REF][START_REF] Qi | Simplification of coefficients in differential equations associated with higher order Frobenius-Euler numbers[END_REF][START_REF] Qi | Simplifying coefficients in differential equations associated with higher order Bernoulli numbers of the second kind[END_REF][START_REF] Qi | Some identities for a sequence of unnamed polynomials connected with the Bell polynomials[END_REF][START_REF] Qi | Simplifying coefficients in differential equations for generating function of Catalan numbers[END_REF][START_REF] Qi | Notes on a family of inhomogeneous linear ordinary differential equations[END_REF][START_REF] Qi | Simplifying and finding ordinary differential equations in terms of the Stirling numbers[END_REF][START_REF] Qi | Simplifying differential equations concerning degenerate Bernoulli and Euler numbers[END_REF][START_REF] Qi | Some properties of the Bernoulli numbers of the second kind and their generating function[END_REF][START_REF] Qi | The inverse of a triangular matrix and several identities of the Catalan numbers[END_REF][START_REF] Zhao | Derivative polynomials of a function related to the Apostol-Euler and Frobenius-Euler numbers[END_REF] are same as the one in this paper.

Remark 6. This paper is a slightly revised version of the preprint [START_REF] Qi | Explicit expressions related to degenerate Cauchy numbers and their generating function[END_REF].

  was first established in [25, Remark 1] and then was applied in [16, Section 2], [17, First proof of Theorem 2], [19, Lemma 2.2], [22, Remark 6.1], [23, Lemma 4], and [32, Lemma 2.6]. The formula (12) and the equivalence were presented in [33, Theorems 2.1 and 4.1].

Acknowledgements. The authors are thankful to Professor Taekyun Kim for his sending a doi link for his paper [2] through an e-mail on 28 February 2018.

The third author was supported by the National Research Foundation of Korea (Grant No. 2018R1D1A1B07041846).