Abdelmejid Bayad
email: abayad@maths.univ-evry.fr

Sébastien Gaboury
email: s1gabour@uqac.ca

FURTHER EXPANSION FORMULAS FOR A CLASS Sébastien Gaboury

S Gaboury

Sebastien Gaboury

FURTHER EXPANSION FORMULAS FOR A CLASS OF GENERALIZED HURWITZ-LERCH ZETA FUNCTIONS OBTAINED BY MEANS OF A NEW LEIBNIZ RULE FOR FRACTIONAL DERIVATIVES

Keywords: 2010 Mathematics Subject Classification. 11M35, 26A33, 33C05, 11B68 Fractional derivatives, expansion formula, Lerch's functional equation, Hurwitz-Lerch zeta functions, Leibniz rule, Riemann zeta function, Generalized Bernoulli polynomials

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Introduction

It is well known that the generalized Hurwitz-Lerch zeta function as well as its extended version have many applications in various areas of mathematics and physics. In number theory, the Riemann and Hurwitz zeta functions are closely related to Dedekind zeta functions and Artin L-functions, which play a central role in the discipline. In addition, the generalized Hurwitz-Lerch zeta functions, evaluated at negative integers, are closely related to the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Frobenius-Euler polynomials [START_REF] Bayad | Reduction and duality of the generalized Hurwitz-Lerch zetas[END_REF][START_REF] Garg | Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions[END_REF][START_REF] Srivastava | Some formulas for the Bernoulli and Euler polynomials at rational arguments[END_REF]. These functions are also connected to the generalized Fermi-Dirac functions and the generalized Bose-Einstein functions [START_REF] Bayad | Reduction and duality of the generalized Hurwitz-Lerch zetas[END_REF]. The generalized Fermi-Dirac and Bose-Einstein functions, which appear in quantum statistics, quantum interference and in the theory of quantum entanglement, have been introduced recently by Srivastava et al. [START_REF] Srivastava | Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions[END_REF]. Moreover, the generalized Hurwitz-Lerch zeta functions have interesting applications in geometric function theory [START_REF] Raducanu | A new class of analytic functions defined by means of a convolution operator involving the Hurwitz-Lerch zeta function[END_REF] and finally, Gupta et al. [START_REF] Gupta | A class of Hurwitz-Lerch zeta distributions and their applications in reliability[END_REF] investigated the generalized Hurwitz-Lerch zeta distribution and applied this new distribution in reliability.

The generalized Hurwitz zeta function ζ(s, a) is defined by [23, p. 88

where

ζ(s, 1) = ζ(s) = 1 2 s -1 ζ s, 1 2 (2)
yields the celebrated Riemann zeta function ζ(s). The Riemann zeta function is continued meromorphically to the whole complex s-plane except for a simple pole at s = 1 with residue 1.

The Hurwitz-Lerch zeta function Φ(z, s, a) is defined, as in [23, p. 121

et seq.], by Φ(z, s, a) := ∞ n=0 z n (n + a) s (3) (a ∈ C \ Z - 0 ; s ∈ C when |z| < 1; Re(s) > 1 when |z| = 1)
. Clearly, we have the following relations

Φ(1, s, a) = ζ(s, a) and Φ(1, s, 1) = ζ(s). (4)
The Hurwitz-Lerch zeta function has the well known integral representation

Φ(z, s, a) = 1 Γ(s) ∞ 0 t s-1 e -at 1 -ze -t dt (5)
(Re(a) > 0; Re(s) > 0 when |z| ≤ 1 (z = 1); Re(s) > 1 when z = 1).

Recently, Lin and Srivastava [START_REF] Lin | Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations[END_REF] investigated a more general family of Hurwitz-Lerch zeta functions. They introduced the function Φ

(ρ,σ) µ,ν (z, s, a) defined by Φ (ρ,σ) µ,ν (z, s, a) := ∞ n=0 (µ) ρn (ν) σn z n (a + n) s (6) µ ∈ C; a, ν ∈ C \ Z - 0 ; ρ, σ ∈ R + ; ρ < σ when s, z ∈ C; ρ = σ and s ∈ C when |z| < 1; ρ = σ and Re(s -µ + ν) > 1 when |z| = 1 ,
where (λ) κ denotes the Pochhammer symbol defined, in terms of the Gamma function, by

(λ) κ := Γ(λ + κ) Γ(λ) = λ(λ + 1) • • • (λ + n -1) (κ = n ∈ N; λ ∈ C) 1 (κ = 0; λ ∈ C \ {0}). (7)
It is easily seen that

Φ (σ,σ) ν,ν (z, s, a) = Φ (0,0) µ,ν (z, s, a) = Φ(z, s, a) (8) and Φ (1,1) µ,1 (z, s, a) = Φ * µ (z, s, a) := ∞ n=0 (µ) n n! z n (n + a) s . ( 9
)
The function Φ * µ (z, s, a) is, in fact, a generalized Hurwitz-Lerch zeta function investigated by Goyal and Laddha [6,p. 100,Equation (1.5)].

Let us recall some other important special cases of the Hurwitz-Lerch zeta function Φ(z, s, a). The Lerch zeta function defined by

l s (ξ) := ∞ n=1 e 2nπiξ n s (ξ ∈ R; Re(s) > 1) (10)
is related to the Hurwitz-Lerch zeta function by the following relation

l s (ξ) = e 2πiξ Φ e 2πiξ , s, 1 . (11)
Also, we note, as a special case of the Hurwitz-Lerch zeta functions, the Lipschitz-Lerch zeta function [23, p. 122, Equation 2.5 [START_REF] Lin | Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions[END_REF]]:

φ(ξ, a, s) := ∞ n=0 e 2nπiξ (n + a) s = Φ e 2πiξ , s, a (12)
(a ∈ C \ Z - 0 ; Re(s) > 0 when ξ ∈ R \ Z; Re(s) > 1 when ξ ∈ Z).
Setting z = exp 2πip q with p ∈ Z and q ∈ N, and using the next series identity

∞ n=0 f (n) = k-1 j=0 ∞ n=0 f (kn + j), (13)
we obtain the following summation formula for the Lipschitz-Lerch zeta function φ(ξ, a, s):

φ p q , a, s = q -s q j=1 ζ s, a + j -1 q exp 2(j -1)pπi q (14)
in terms of the Hurwitz zeta function ζ(s, a) defined by equation (1) (see also [21, p. 81, Equation (3.9)]).

A very important result for the sequel is the Lerch's functional equation [3, p. 29, Equation 1.11 [START_REF] Gupta | A class of Hurwitz-Lerch zeta distributions and their applications in reliability[END_REF]]:

Φ(z, s, a) = i z -a (2π) s-1 Γ(1 -s) exp - 1 2 sπi Φ e -2πia , 1 -s, log z 2πi -exp 2a + 1 2 s πi Φ e 2πia , 1 -s, 1 - log z 2πi (15) Re(s) < 0; |arg(-log(z) mod 2πi)| ≤ π; a ∈ C \ Z - 0 .
Recently, Lin et al. [START_REF] Lin | Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions[END_REF] obtained some expansion formulas for a generalized Hurwitz-Lerch zeta function Φ * µ (z, s, a) defined by [START_REF]Fundamental properties of fractional derivatives via Pochhammer integrals[END_REF]

D µ z f (z)g(z) = ∞ j=0 µ j D µ-j z f (z)D j z g(z) (16)
where D µ z denotes the Riemann-Liouville fractional calculus operator [4, p. 181 et seq.] defined by:

D µ z f (z) :=        1 Γ(-µ) z 0 f (t)(z -t) -µ-1 dt if Re(µ) < 0, d m dz m D µ-m z f (z) if m -1 ≤ Re(µ) < m. (17)
Explicitly, they proved the following expansion formula for the generalized Hurwitz-Lerch zeta function Φ * µ (z, s, a):

Φ * µ (z, s, a) = i z -a Γ(1 -s) ∞ j=0 (j -a + 1) µ-j-1 Γ(µ -j)j! j k=0 j -1 k -1 (1 -s) k B (j) j-k • (2π) s-k-1 exp - 1 2 (s -k)πi Φ e -2πia , 1 -s + k, log z 2πi -exp 2a + 1 2 (s -k) πi Φ e 2πia , 1 -s + k, 1 - log z 2πi (18) with Re(µ) > 0, s ∈ C, |arg(-log(z) mod 2πi)| ≤ π, a ∈ C \ Z - 0 , where B (α) n := B (α)
n (0) are the generalized Bernoulli numbers [START_REF] Luke | The special functions and their approximations[END_REF] given by the following generating function:

z e z -1 α e xz = ∞ n=0 B (α) n (x) z n n! (|z| < 2π; 1 α := 1). ( 19
)
The aim of this paper is to make use of a new generalized Leibniz rule for fractional derivatives obtained recently by Tremblay et al. [START_REF] Tremblay | A new Leibniz rule and its integral analogue for fractional derivatives[END_REF] by means of a representation based on the Pochhammer's contour of integration for fractional derivatives [START_REF] Lavoie | Fundamental properties of fractional derivatives via Pochhammer integrals[END_REF] in order to derive a new expansion formula for the generalized Hurwitz-Lerch zeta function Φ * µ (z, s, a) in terms of the Hurwitz-Lerch zeta function Φ(z, s, a). Moreover, an expansion theorem is obtained for the generalized family of Hurwitz-Lerch zeta functions Φ (ρ,σ) µ,ν (z, s, a). Special cases are also given. The paper is arranged as follows. Section 2 is devoted to the representation of the fractional derivatives based on the Pochhammer's contour of integration and the new generalized Leibniz rule. In section 3, we prove the expansions formulas and finally, section 4 aims to provide some special cases.

Pochhammer contour integral representation for fractional derivative and a new generalized Leibniz rule

The use of contour of integration in the complex plane provides a very powerful tool in both classical and fractional calculus. The most familiar representation for fractional derivative of order α of z p f (z) is the Riemann-Liouville integral [START_REF] Erdelyi | An integral equation involving Legendre polynomials[END_REF][START_REF] Liouville | Mémoire sur le calcul des différentielles à indices quelconques[END_REF][START_REF] Riesz | L'intégrale de Riemann-Liouville et le problème de Cauchy[END_REF] that is

D α z z p f (z) = 1 Γ(-α) z 0 f (ξ)ξ p (ξ -z) -α-1 dξ, (20)
which is valid for Re(α) < 0, Re(p) > 1. Here the integration is done along a straight line from 0 to z in the ξ-plane. By integrating by part m times, we obtain

D α z z p f (z) = d m dz m D α-m z z p f (z). ( 21
)
This allows to relax the restriction Re(α) < 0 by Re(α) < m [START_REF] Riesz | L'intégrale de Riemann-Liouville et le problème de Cauchy[END_REF]. Another used representation for the fractional derivative is the one based on the Cauchy integral formula widely used by Osler [START_REF] Osler | Fractional derivatives of a composite function[END_REF][START_REF]Leibniz rule for the fractional derivatives and an application to infinite series[END_REF][START_REF]Leibniz rule, the chain rule and Taylor's theorem for fractional derivatives[END_REF][START_REF]Fractional derivatives Leibniz rule[END_REF]. These two representations have been used in many interesting research papers. It appears that the less restrictive representation of fractional derivative according to parameters is the Pochhammer's contour definition introduced in [START_REF]Fundamental properties of fractional derivatives via Pochhammer integrals[END_REF][START_REF] Tremblay | Une contribution à la théorie de la dérivée fractionnaire[END_REF]. Definition 2.1. Let f (z) be analytic in a simply connected region R. Let g(z)be regular and univalent on R and let g -1 (0) be an interior point of R. If α is not a negative integer, p is not an integer, and z is in R -{g -1 (0)}, we define the fractional derivative of order α of g(z) p f (z) with respect to g(z) by

D α g(z) g(z) p f (z) = e -iπp Γ(1 + α) 4π sin(πp) C(z+,g -1 (0)+,z-,g -1 (0)-;F (a),F (a)) f (ξ)g(ξ) p g (ξ) (g(ξ) -g(z)) α+1 dξ (22)
For non-integer α and p, the functions g(ξ) p and (g(ξ) -g(z)) -α-1 in the integrand have two branch lines which begin respectively at ξ = z and ξ = g -1 (0), and both pass through the point ξ = a without crossing the Pochhammer contour P (a) = {C 1 ∪ C 2 ∪ C 3 ∪ C 4 } at any other point as shown in Figure 1. F (a) denotes the principal value of the integrand in [START_REF] Srivastava | Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions[END_REF] at the beginning and ending point of the Pochhammer contour P (a) which is closed on Riemann surface of the multiple-valued function F (ξ).

Remark 2.2. In the Definition 2.1, the function f (z) must be analytic at ξ = g -1 (0). However it is interesting to note here that we could also allow f (z) to have an essential singularity at ξ = g -1 (0), and the equation [START_REF] Srivastava | Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions[END_REF] would still be valid.

Remark 2.3. The Pochhammer contour never crosses the singularities at ξ = g -1 (0) and ξ = z in [START_REF] Srivastava | Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions[END_REF]. Then we know that the integral is analytic for all p and α and for z in R -{g -1 (0)}. Indeed, the only possible singularities of D α g(z) g(z) p f (z) are α = -1, -2, . . . , and p = 0, ±1, ±2, . . . which can directly be identified from the coefficient of the integral [START_REF] Srivastava | Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions[END_REF]. However, integrating by parts N times the integral in [START_REF] Srivastava | Some extensions of the Fermi-Dirac and Bose-Einstein functions with applications to the family of the zeta and related functions[END_REF] by two different ways, we can show that α = -1, -2, . . . , and p = 0, 1, 2, . . . are removable singularities (see [START_REF]Fundamental properties of fractional derivatives via Pochhammer integrals[END_REF]) .

D α z z p = Γ(1 + p) Γ(1 + p -α) z p-α (Re(p) > -1), (23)
but by adopting the Pochhammer based representation for the fractional derivative this last restriction becomes p not a negative integer. In view of definition 6, the fractional derivative formula for the generalized Hurwitz-Lerch zeta function Φ (ρ,σ) µ,ν (z, s, a) with ρ = σ [10, p. 730, Equation [START_REF] Tremblay | Une contribution à la théorie de la dérivée fractionnaire[END_REF]] is

D µ-ν z z µ-1 Φ(z σ , s, a) = Γ(µ) Γ(ν) z ν-1 Φ (σ,σ) µ,ν (z σ , s, a) (24)
with µ -1 not a negative integer.

A very interesting special case is obtained when setting ν = σ = 1, and equation [START_REF] Tremblay | Une contribution à la théorie de la dérivée fractionnaire[END_REF] reduces to the following form:

Φ * µ (z, s, a) = 1 Γ(µ) D µ-1 z z µ-1 Φ(z, s, a) (25)
with µ -1 not a negative integer.

As remarked by Lin and Srivastava [START_REF] Lin | Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations[END_REF], the function Φ * µ (z, s, a) is essentially a fractional derivative of the classical Hurwitz-Lerch function Φ(z, s, a). Many other interesting explicit representations for Φ * µ (z, s, a) have been proven by Lin and Srivastava [START_REF] Lin | Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations[END_REF].

Recently, one of the authors obtained a new generalized Leibniz rule for fractional derivatives by making use of the properties of this representation [START_REF] Tremblay | A new Leibniz rule and its integral analogue for fractional derivatives[END_REF][START_REF]Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives[END_REF]. Explicitly, we proved the following theorem: Theorem 2.4. (i) Let R be a simply connected region containing the origin. (ii) Let u(z) and v(z) satisfy the conditions of Definition 2.1 for the existence of the fractional derivative. (iii) Let U ⊂ R be the region of analyticity of the function u(z) and V ⊂ R the one for the function v(z). Then for z = 0, z ∈ U ∩ V, Re(1 -β) > 0 the following product rule holds (26)

D α z z α+β-1 u(z)v(z) = z sin(βπ)Γ(1 + α) sin(µπ) sin((α + β -µ)π) sin((α + β)π) sin((β -µ -ν)π) sin((µ + ν)π) • ∞ n=-∞ D α+ν+1-n z z α+β-µ-1-n u(z)D -1-ν+n z z µ-1+n v(z) Γ(2 + α + ν -n)Γ(-ν + n) .

Expansion formulas

In this section, we present the two expansion formulas obtained by means of fractional calculus techniques for the functions Φ (ρ,σ) µ,ν (z, s, a) and Φ * µ (z, s, a). The first expansion formula is a direct applications of the generalized Leibniz rule [START_REF]Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives[END_REF]. The second expansion formula is analogous to Lerch's functional equation. We first state the first one in the form of the theorem below: Theorem 3.1. The following expansion formula holds true

Φ (σ,σ) µ,ν (z σ , s, a) = Γ(ν)Γ(µ -ν + 1) sin βπ sin(µ -ν + β -λ)π Γ(µ)Γ(ν -γ -λ -1)Γ(1 + γ + λ) sin(µ -ν + β)π sin(β -λ -γ)π • sin λπ sin(λ + γ)π ∞ n=-∞ Γ(µ -λ -n)Γ(λ + n) Γ(2 + µ -ν + γ -n)Γ(-γ + n) Φ (σ,σ)
λ+n,1+γ+λ (z σ , s, a)

(27) z = 0; Re(1-β) > 0; (λ+n) ∈ C\Z - 0 ; (1+λ+γ) ∈ C\Z - 0 ; a, ν, µ ∈ C\Z - 0 ; σ ∈ R + ; s ∈ C when |z σ | < 1 .
Proof. First, making the substitutions µ → λ, ν → γ in Theorem 2.4 and afterwards setting α = µ -ν, u(z) = z ν-β and v(z) = Φ (z σ , s, a) (both u(z) and v(z) satisfied the conditions of Theorem 2.4), we have

D µ-ν z z µ-1 Φ (z σ , s, a) = z Γ(µ -ν + 1) sin βπ sin λπ sin(µ -ν + β -λ)π sin(µ -ν + β)π sin(β -λ -γ)π sin(λ + γ)π • ∞ n=-∞ D µ-ν+γ+1-n z z µ-λ-1-n D -1-γ+n z z λ-1+n Φ (z σ , s, a) Γ(2 + µ -ν + γ -n)Γ(-γ + n) . ( 28
)
Using ( 23) and ( 24), we find that

D µ-ν+γ+1-n z z µ-λ-1-n = Γ(µ -λ -n) Γ(ν -γ -λ -1) z ν-γ-λ-2 (29) and D -1-γ+n z z λ-1+n Φ (z σ , s, a) = Γ(λ + n) Γ(γ + λ + 1) z γ+λ Φ (σ,σ)
λ+n,γ+λ+1 (z σ , s, a) . Finally, the results follow by combining (29), ( 30) and (28). Now let us shift our focus on the second expansion formula involving the generalizerd Hurwitz-Lerch zeta function Φ * µ (z, s, a) which is stated in the next theorem: Theorem 3.2. The following expansion formula holds true for Φ * µ (z, s, a):

Φ * µ (z, s, a) = i Γ(1 -s)z -a sin βπ sin(β + µ -ω)π Γ(µ)Γ(1 -ω -a) sin(µ + β)π sin(β -ω)π ∞ n=0 n k=0 k j=0 n -1 k -1 • k j Γ(µ -a -ω -n) Γ(µ -n)n! B (n) n-k (2π) s-1-j (1 -s) j (ω -1 + n) k-j • exp - (s -j)πi 2 Φ e -2πia , 1 -s + j, log z 2πi -exp 2a + s -j 2 πi Φ e 2πia , 1 -s + j, 1 - log z 2πi (31) µ ∈ C \ Z - 0 ; ω ∈ C; Re(1 -β) > 0; s ∈ C; |arg(-log(z) mod 2πi)| ≤ π; a ∈ C \ Z - 0 .
Proof. Multiplying each member of equation ( 15) by z µ-1 and operating both sides of the resulting equation by D µ-1 z , we get

D µ-1 z z µ-1 Φ(z, s, a) = i (2π) s-1 Γ(1 -s) exp - iπs 2 D µ-1 z z µ-1-a Φ e -2πia , 1 -s, log z 2πi -exp 2a + s 2 iπ D µ-1 z z µ-1-a Φ e 2πia , 1 -s, 1 - log z 2πi . (32)
Using (25), we have

Φ * µ (z, s, a) = i (2π) s-1 Γ(1 -s) Γ(µ) exp - iπs 2 D µ-1 z z µ-1-a Φ e -2πia , 1 -s, log z 2πi -exp 2a + s 2 iπ D µ-1 z z µ-1-a Φ e 2πia , 1 -s, 1 - log z 2πi . (33)
We now have to evaluate the two fractional derivatives on the right hand side of (33). By applying the Leibniz rule ( 26) when ν = -1 on each member of the right hand side of (32) with u(z) = z 1-β-a and v(z) = Φ e -2πia , 1 -s, log z 2πi in the first member, and with u(z) = z 1-β-a and

v(z) = Φ e 2πia , 1 -s, 1 - log z 2πi
in the second member (we see that the functions u(z) and v(z) involved satisfied the conditions of Theorem 2.4), we obtain respectively:

D µ-1 z z µ-1-a Φ e -2πia , 1 -s, log z 2πi = z Γ(µ) sin βπ sin(µ + β -ω)π sin(µ + β)π sin(β -ω)π • ∞ n=0 D µ-1-n z z µ-ω-a-1-n D n z z ω-1+n Φ e -2πia , 1 -s, log z 2πi Γ(µ -n)n! (34)
and

D µ-1 z z µ-1-a Φ e 2πia , 1 -s, 1 - log z 2πi = z Γ(µ) sin βπ sin(µ + β -ω)π sin(µ + β)π sin(β -ω)π • ∞ n=0 D µ-1-n z z µ-ω-a-1-n D n z z ω-1+n Φ e 2πia , 1 -s, 1 -log z 2πi Γ(µ -n)n! . (35)
Equations ( 34) and (35) reduce after simple calculations to:

D µ-1 z z µ-1-a Φ e -2πia , 1 -s, log z 2πi = z 1-ω-a Γ(µ) sin βπ sin(µ + β -ω)π Γ(1 -ω -a) sin(µ + β)π sin(β -ω)π ∞ n=0 Γ(µ -ω -a -n)z -n Γ(µ -n) n! • (z n D n z ) z ω-1+n Φ e -2πia , 1 -s, log z 2πi (36) and D µ-1 z z µ-1-a Φ e 2πia , 1 -s, 1 - log z 2πi = z 1-ω-a Γ(µ) sin βπ sin(µ + β -ω)π Γ(1 -ω -a) sin(µ + β)π sin(β -ω)π ∞ n=0 Γ(µ -ω -a -n)z -n Γ(µ -n) n! • (z n D n z ) z ω-1+n Φ e 2πia , 1 -s, 1 - log z 2πi . ( 37
)
We know from [13, p. 24, Equations ( 5) and ( 6)] that 36) and (37) become respectively

z n D n z = n k=0 n -1 k -1 B (n) n-k (zD z ) k . ( 38
D µ-1 z z µ-1-a Φ e -2πia , 1 -s, log z 2πi = z 1-ω-a Γ(µ) sin βπ sin(µ + β -ω)π Γ(1 -ω -a) sin(µ + β)π sin(β -ω)π ∞ n=0 n k=0 n -1 k -1 Γ(µ -ω -a -n)z -n Γ(µ -n) n! • B (n) n-k D x 2πi k e 2πix(ω-1+n) Φ e -2πia , 1 -s, x (41)
and

D µ-1 z z µ-1-a Φ e 2πia , 1 -s, 1 - log z 2πi = z 1-ω-a Γ(µ) sin βπ sin(µ + β -ω)π Γ(1 -ω -a) sin(µ + β)π sin(β -ω)π ∞ n=0 n k=0 n -1 k -1 Γ(µ -ω -a -n)z -n Γ(µ -n) n! • B (n) n-k -D y 2πi k e -2πi(y-1)(ω-1+n) Φ e 2πia , 1 -s, y . (42)
With the help of the classical Leibniz rule and the fact that

∂ k ∂a k Φ(z, s, a) = (-1) k (s) k Φ(z, s + k, a), (43)
the result follows after some simple calculations.

Special cases of Theorems 3.1 and 3.2

In this section, we first give two special cases of Theorem 3.1 involving the Hurwitz-Lerch zeta function Φ * µ (z, s, a). Next, we shift our focus on some interesting consequences of Theorem 3.2.

If we put σ = 1, ν = 1 and γ = -1 in equation ( 27) and use equation ( 9), we find

Φ * µ (z, s, a) = sin βπ sin(µ + β -λ)π Γ(1 -λ)Γ(λ) sin(µ + β)π sin(β -λ)π • ∞ n=0 Γ(µ -λ -n)Γ(λ + n) Γ(µ -n) n! Φ (1,1) λ+n,λ (z, s, a) (44) z = 0; Re(1-β) > 0; (λ+n) ∈ C\Z - 0 ; a, µ ∈ C\Z - 0 ; s ∈ C when |z| < 1 .
Setting σ = 1, µ = ν in equation ( 27) and with the help of equation ( 8), we obtain the following expansion formula:

Φ(z, s, a) = sin λπ sin(β -λ)π Γ(ν -γ -λ -1)Γ(1 + γ + λ) sin(β -λ -γ)π sin(λ + γ)π • ∞ n=-∞ Γ(ν -λ -n)Γ(λ + n) Γ(2 + γ -n)Γ(-γ + n) Φ (1,1) λ+n,1+γ+λ (z, s, a) (45) z = 0; Re(1 -β) > 0; (λ + n) ∈ C \ Z - 0 ; (1 +λ + γ) ∈ C \ Z - 0 ; a, ν ∈ C \ Z - 0 ; s ∈ C when |z| < 1 .
Let us now shift our focus on some interesting consequences of Theorem 3.2. Let z = e 2πiξ and a = p q (p ∈ Z; q ∈ N; ξ ∈ R)

and replace s by µ -s, and then apply definition [START_REF] Liouville | Mémoire sur le calcul des différentielles à indices quelconques[END_REF] and the series identity [START_REF] Miller | An introduction of the fractional calculus and fractional differential equations[END_REF], we obtain from our theorem:

Φ * µ e 2πiξ , µ -s, p q = i Γ(1 -µ + s) sin βπ sin(β + µ -ω)π Γ(µ)Γ 1 -ω -p q sin(µ + β)π sin(β -ω)π ∞ n=0 n k=0 k j=0 n -1 k -1 k j • Γ µ -p q -ω -n Γ(µ -n)n! B (n) n-k (2πq) µ-s-1-j (1 -µ + s) j (ω -1 + n) k-j • q r=1 ζ 1 -µ + s + j, ξ + r -1 q exp -iπ µ -s -j 2 + 2(r -1 + ξ)p q - q r=1 ζ 1 -µ + s + j, r -ξ q exp iπ µ -s -j 2 + 2(r -ξ)p q (46) µ ∈ C \ Z - 0 ; ω ∈ C; Re(1 -β) > 0; s ∈ C; 0 < ξ < 1; a ∈ C \ Z - 0 .
Moreover, if we set µ = m (m ∈ N) in equations ( 31

k j • Γ m -p q -ω -n Γ(m -n)n! B (n) n-k (2πq) m-s-1-j (1 -m + s) j (ω -1 + n) k-j • q r=1 ζ 1 -m + s + j, ξ + r -1 q exp -iπ m -s -j 2 + 2(r -1 + ξ)p q - q r=1
ζ 1 -m + s + j, r -ξ q exp iπ m -s -j 2 + 2(r -ξ)p q (48)

ω ∈ C; Re(1 -β) > 0; s ∈ C; 0 < ξ < 1; a ∈ C \ Z - 0 .
Setting m = 1 in (47) yields the Lerch's functional equation [START_REF] Osler | Fractional derivatives of a composite function[END_REF].

For the sake of completeness, we choose to recall the expansion formulas analogue to equation ( 47

  a) s (Re(s) > 1; a ∈ C \ Z - 0 ; Z - 0 := {0, -1, -2, ...})

Figure 1 .

 1 Figure 1. Pochhammer's contour It is well known that [14, p. 83, Equation (2.4)]

  to mention that in the case where ρ = σ and |z σ | = 1, the function Φ (ρ,σ) µ,ν (z, s, a) exists for Re(s -µ + ν) > 1. This implies that for |z σ | = 1, the function Φ (σ,σ) λ+n,γ+λ+1 (z σ , s, a) in (30) must satisfy Re(s + γn) > 0 for all n ∈ Z which case is impossible. Thus, the expansion holds true only for |z σ | < 1.

  ) and (46), we obtain respectivelyΦ * m (z, s, a) = i Γ(1 -s)z -a Γ(m)Γ(1 -ω -a) -a -ω -n) Γ(m -n)n! B (n) n-k (2π) s-1-j (1 -s) j (ω -1 + n) k-j • exp -(s -j)πi 2 Φ e -2πia , 1 -s + j, log z 2πi -exp 2a + s -j2πi Φ e 2πia , 1 -s + j, 1 -

•

  ) obtained recently by Lin et al.[START_REF] Lin | Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions[END_REF] p. 825, Equation (40)] and Garg et al.[START_REF] Garg | Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions[END_REF] Theorem 2]. Respectively, they obtained:Φ * m (z, s, a) = i Γ(1 -s)z -a m-1 j=0 (j -a + 1) m-j-1 j! (m -j -(2π) s-1-k exp -(s -k)πi 2 Φ e -2πia , 1 -s + k, log z 2πi -exp 2a + s -k 2 πi Φ e 2πia , 1 -s + k, 1 -) k-j (2π) s-j-1 exp -(s -j)πi 2 Φ e -2πia , 1 -s + j, log z 2πi -exp 2a + s -j 2 πi Φ e 2πia , 1 -s + j, 1 -log z 2πi(50)with s ∈ C, |arg(-log(z) mod 2πi)| ≤ π, a ∈ C \ Z - 0 .


I Shestopalova
email: iuliia.shestopalova@tuni.fi

J Korri

A Lemiere

M Närhi

R Gumenyuk

C Boussard-Plédel

J Troles

L Petit

Addition of Ag 2 O in Er 3+ doped oxyfluorophosphate glass to allow the drawing of optical fibers

Here, Ag 2 O containing glasses in the NaPO 3 -CaF 2 glass network were prepared using standard melting process. The addition of Ag 2 O was found to increase the thermal stability of the glass due to the decrease in the Q 2 units at the expense of Q 1 units, to decrease the intensity of the upconversion under 980 nm pumping and to have a small impact on the nucleation and growth mechanism. Due to the thermal stability against crystallization of the glass prepared with 4 mol % of Ag 2 O, we demonstrate that fiber can be drawn from this glass. Despite the formation of Ag nanoparticles at the surface of the fiber although the drawing is a fast process, light can still be confined in the fiber. The fiber exhibits a large emission band centered at 1.5 μm under 980 nm pumping.

Introduction

For the past decades, Er 3+ doped optical fibers have been of great interest as they can be used not only in photonics applications (telecommunications, solar panels, etc.) but also in biomedical applications (bio-imaging, bio-sensing, etc.) [START_REF] Keiser | Review of diverse optical fibers used in biomedical research and clinical practice[END_REF][START_REF] Zhong | A review on nanostructured glass ceramics for promising application in optical thermometry[END_REF][START_REF] Florêncio | Efficiency enhancement in solar cells using photon down-conversion in Tb/Yb-doped tellurite glass[END_REF][START_REF] Tünnermann | Fiber lasers and amplifiers: an ultrafast performance evolution[END_REF]. Although Er 3+ doped silica and silicate glasses have been intensively investigated, their main limitation is their low Er 3+ solubility [START_REF] Gapontsev | Erbium glass lasers and their applications[END_REF]. Thus, effort has been focused worldwide on the development of new glass compositions with enhanced spectroscopic properties (increased lifetime and larger absorption and emission cross-sections). When developing new glasses to be drawn into optical fibers, it is crucial that the new glasses possess proper thermal properties. Indeed, the glasses should be thermally stable to allow the drawing of optical fiber without crystallizing the amorphous network.

Phosphate glasses are known to have high solubility of rare-earth ions (RE). The luminescence quenching in these glasses occurs at high RE content [START_REF] Boetti | High concentration Yb-Er co-doped phosphate glass for optical fiber amplification[END_REF] compared to silica and silicate glasses [START_REF] Boetti | High concentration Yb-Er co-doped phosphate glass for optical fiber amplification[END_REF]. As a consequence, when doped with Er 3+ ions, phosphate glasses are promising as amplifiers and wavelength division multiplexer operating at 1.5 μm [START_REF] Hwang | Erbium-doped phosphate glass fibre amplifiers with gain per unit length of 2.1 dB/cm[END_REF].

They can also be used as upconversion (UC) lasers (conversion of the IR photons to visible photons) [START_REF] Feng | Optical properties and upconversion in rare earth doped oxyfluoride glasses[END_REF], just to cite an example. Recently, the Er 3+ doped glass with the composition 75NaPO 3 -25CaF 2 (in mol%) was reported to be promising, especially for upconversion application. Indeed, intense green and red emissions were observed under NIR pumping after thermally treating the glass to induce the volume precipitation of CaF 2 crystals [START_REF] Nommeots-Nomm | Luminescence of Er 3+ doped oxyfluoride phosphate glasses and glass-ceramics[END_REF]. However, due to the small temperature difference between the crystallization and glass transition temperatures (~50 • C), fiber can not be drawn from this glass preform. The composition of the phosphate glasses can be easily tailored [START_REF] Bunker | Phosphate glass dissolution in aqueous solutions[END_REF]. However, the change in the glass composition might impact the spectroscopic properties and more importantly the nucleation and growth mechanism as reported recently in Ref. [START_REF] Ojha | Tailoring the glass composition to increase the thermal stability without impacting the crystallization behavior of oxyfluorophosphate glass[END_REF]. The thermal stability of the Er 3+ doped 75NaPO 3 -25CaF 2 glass was reported to increase when adding MgO, Al 2 O 3 or Fe 2 O 3 . However, the changes in the glass composition depolymerize the glass network limiting the precipitation of CaF 2 crystals and thus leading to lower intensity of upconversion under NIR pumping compared to the Er 3+ doped 75NaPO 3 -25CaF 2 glass after thermal treatment.

Here, the impact of adding Ag 2 O in the Er 3+ glass with the composition 75NaPO 3 -25CaF 2 (in mol%) on the thermal properties is investigated. Ag 2 O was selected as it is well known that Ag nanoparticles (NPs) can precipitate during thermal treatment. Due to the surface plasmon resonance (SPR) of Ag NPs, the spectroscopic properties of Er 3+ doped glass could be enhanced [START_REF] Fares | Silver nanoparticles enhanced luminescence properties of Er 3+ doped tellurite glasses: effect of heat treatment[END_REF][START_REF] Kuusela | Impact of ZnO addition on Er 3+ nearinfrared emission, the formation of Ag nanoparticles, and the crystallization of sodium fluorophosphate glass[END_REF].

Experimental

Phosphate glasses with the composition (in mol%) (98-x) (0.75NaPO 3 -0.25CaF 2 ) -xAg 2 O -2ErF 3 with x = 0, 0.5, 1, 2 and 4 (in mol%) were prepared using standard melting quenching technique. The glasses were labeled as Agx. NaPO 3 (Alfa Aesar, tech.), CaF 2 (Alfa Aesar, 99.5%), Ag 2 SO 4 (Sigma Aldrich, ≥ 99.5%) and ErF 3 (Alfa Aesar, 99.99%) were used as the raw materials. The 15g batches were melted in alumina crucible at 1050 • C. After quenching, the glasses were annealed at 40 • C below their respective glass transition temperature (T g ) for 5 h to release the stress from the quench. The 9 cm long preform with 1 cm diameter was prepared from the Ag4 glass and was drawn into fiber with a diameter of ~325 μm using a drawing temperature of 455 • C under helium gas with a flow of 1.5 L/min so the preform is in an inert atmosphere. No coating was used during the drawing to allow the measurement of the thermal and structural properties of the fibers. However, the fibers were fragile and so few meters of fiber were obtained.

Archimedes' method was used to determine the density of the glasses. Densities are given with an accuracy of ±0.02 g/cm 3 .

The thermal properties of the glasses were measured using differential thermal analysis (DTA, Netszch JUPITER F1 instrument). The glass transition temperature (T g ) was taken as the inflection point of the endotherm peak, T x at the onset of the crystallization peak and the crystallization temperature (T p ) at the maximum of the exothermic peak. The accuracy of measurement is ±3 • C.

The X-ray diffraction (XRD) analyzer Philips X'pert (Philips, Amsterdam, Netherlands) with Cu Kα X-ray radiation (λ = 1.5418 Å) was used to identify the crystalline phases precipitating in the glasses. A "zero-background holder" Fe plate was used as sample holder which was rotated around the Φ-axis at a 16 revolutions per minute. Data were collected with a step size of 0.003 • .

A scanning electron microscope (SEM) (Crossbeam 540, Carl Zeiss, Oberkochen, Germany) coupled to an EDS detector (X-MaxN 80, Oxford Instruments, Abingdon-on-Thames, UK) were used to analyze the samples.

The Vis-NIR spectrophotometer (UV-3600 Plus, Shimadzu) was used to measure the absorption and transmittance spectra of the glasses. From the absorption coefficient, the absorption cross-section σ abs (λ) was calculated using Equation (1):

σ abs (λ) = 2.303 log(I 0 /I) L × N Er 3+ (1)
where log (I 0 /I) is the absorbance, L is the thickness of the sample (in cm) and N Er 3+ is the concentration of rare-earth ions (ions/cm 3 ). The

Table 1

Physical and thermal properties of the investigated glasses. accuracy of absorption cross-section is ±10%.

Glass T g ( • C) ± 3 • C T x ( • C) ± 3 • C T p ( • C) ± 3 • C ΔT = T x -T g ± 6 • C ρ (g/cm -3 ) ± 0.
The upconversion (UC) spectra were collected from powder at room temperature using a Spectro 320-131 optical spectrum analyzer (OSA, Instrument Systems Optische Messtechnik GmbH). A continuous-wave 980 nm monochromatic single-mode fiber pigtailed laser diode (CM962UF76P-10R, Oclaro) was used for excitation using 1 A.

The PerkinElmer Spectrum FTIR2000 spectrometer in attenuated total reflectance (ATR) mode was used to measure the IR spectra of the glasses with a resolution of 2 cm -1 and 8 scan accumulation.

The spectral transmission and emission of the optical fiber were measured in the 6, 3 or 1.5 cm-long tested fiber samples by launching a broadband light from the fiber-pigtailed white light source (MG922A, Anritsu) or a fiber-pigtailed 980 nm laser source (Coherent), respectively. The white light was used to measure the spectral transmission of the fiber from 600 nm to 1750 nm and the 980 nm laser source was used to measure the emission centered at 1.55 μm. Both light sources were launched into samples by butt-coupling technique via a multimode fiber. The output light was then collected into another multimode fiber and analyzed by an OSA (ANDO AQ6317B).

Results and discussion

Er 3+ doped glasses with the 75NaPO 3 -25CaF 2 composition (in mol %) were prepared with different amounts of Ag 2 O (x). As depicted in Table 1, an increase in x increases the density, T x and T p while it decreases T g . As a consequence, ΔT, used to evaluate the thermal stability of the glass against crystallization [START_REF] Çelikbilek | Investigation on thermal and microstructural characterization of the TeO 2 -WO 3 system[END_REF], increases when adding Ag 2 O in the phosphate glass clearly evidencing that Ag 2 O can be used to enhance the thermal stability of the oxyfluorophosphate glass. The Ag1, Ag2 and Ag4 glasses possess a ΔT >90 • C (considering the accuracy of ± 6 • C) suggesting that no crystallization would occur during the fiber drawing from preform. The increase in density with the progressive addition of In order to understand the thermal properties of the Ag containing glasses, the structure of the glasses was investigated using FTIR. As shown in Fig. 1a, the IR spectra are similar to those reported in Refs. [START_REF] Nommeots-Nomm | Luminescence of Er 3+ doped oxyfluoride phosphate glasses and glass-ceramics[END_REF][START_REF] Ojha | Nucleation and growth behavior of Er 3+ doped oxyfluorophosphate glasses[END_REF]. A complete attribution of the IR bands can be found in Refs. [START_REF] Nommeots-Nomm | Luminescence of Er 3+ doped oxyfluoride phosphate glasses and glass-ceramics[END_REF][START_REF] Ojha | Nucleation and growth behavior of Er 3+ doped oxyfluorophosphate glasses[END_REF]. The progressive addition of Ag 2 O leads to slight changes in the shape of the IR spectrum. An increase in x leads to a slight decrease in intensity of the band at 1250 cm -1 compared to the main band, the position of which shifts to lower wavelength. These changes in the IR spectra suggest an increase in the Q 1 units (increase in intensity of the band at ~980 cm -1 ) at the expense of Q 2 units (decrease in intensity of the band at ~1260 cm -1 ) when adding Ag 2 O in the glass. Therefore, the addition of Ag 2 O is thought to depolymerize the phosphate network. Similar changes in the IR spectra were reported when adding other elements such as Al 2 O 3 , MgO and Fe 2 O 3 [START_REF] Ojha | Tailoring the glass composition to increase the thermal stability without impacting the crystallization behavior of oxyfluorophosphate glass[END_REF]. The depolymerization of the network is suspected to increase slightly the content of OH groups as evidenced by the increase in intensity of the band located in the 2600-3700 cm -1 range, attributed to strongly (2800 cm -1 ) and weakly (3500 cm -1 ) associated hydroxyl groups [START_REF] Ohkawa | Influence of water on non-radiative decay of Yb 3 + -2 F 5/2 level in phosphate glass[END_REF] (Fig. 1b).

The changes in the glass structure are also confirmed from the shift of the optical band gap to longer wavelength when x increases (Fig. 1c). The typical absorption bands of Er 3+ are clearly visible in the absorption spectra of the glasses. They correspond to the transitions from the ground state ( 4 I 15/2 ) to the various excited levels of Er 3+ ions. Ag + cations are expected to be dispersed in the network because of the absence of the SPR absorption band of Ag NPs usually located at ~400 nm [START_REF] Gangopadhyay | Growth of silver nanoclusters embedded in soda glass matrix[END_REF]. The glasses exhibit similar absorption cross-sections at 980 nm and 1530 nm at (1.6 × 10 -21 ) and (5.3 × 10 -21 ) cm 2 , independently of their composition. These absorption cross-section values are similar to those reported for phosphate glasses [START_REF] Ennouri | Investigations of the thermal, structural, and Near-IR emission properties of Ag containing fluorophosphate glasses and their crystallization process[END_REF]. The shape of the absorption bands remains unchanged after adding Ag 2 O in the glass. From the optical properties of the glasses, the site of Er 3+ ions are suspected to be similar in the investigated glasses.

As shown in Fig. 2a, the addition of Ag 2 O in the oxyfluorophosphate network has an impact on the upconversion (UC) properties of the glasses. The UC spectra of the glasses exhibit the typical green (525 and 550 nm) and red (650 nm) emission bands of Er 3+ ions, which are due to the 2 H 11/2 , 4 S 3/2 → 4 I 15/2 and 4 F 9/2 → 4 I 15/2 transition of Er 3+ ions, respectively. According to Ref. [START_REF] Balda | Infrared-to-visible upconversion of Er 3+ ions in GeO 2 -PbO-Nb 2 O 5 glasses[END_REF], the energy-transfer upconversion and excited-state absorption are thought to be responsible for the upconversion luminescence. An increase in x decreases the intensity of the green emission while increasing the intensity of the red emission. It also decreases significantly the integrated upconversion emission area between 500 and 700 nm (Fig. 2b). One should mention that the high overall intensity of emission for the glass with x = 4 (Ag4) compared to the other Ag containing glasses is due to the large intensity of the red emission. The changes in the UC properties when adding Ag 2 O in the network suggest modification in the local symmetry of Er 3+ ions when adding Ag 2 O in the glass. However, as the glasses have similar absorption coefficient at 980 nm, the decrease in the intensity of the upconversion can be related to the depolymerization of the phosphate network induced by the addition of Ag 2 O and also to the small difference in the content in the OH groups, known as one of the main detrimental factors affecting the upconversion (UC) efficiency [START_REF] Feng | Internal OH-induced cascade quenching of upconversion luminescence in NaYF 4 :Yb/Er nanocrystals[END_REF]. It is also possible that the red emission of Er 3+ ions is related to a local increase in the concentration of Er 3+ ions when adding Ag 2 O. The high concentration of Er 3+ is suspected to lead to the 4 S 3/2 + 4 I 9/2 → 4 F 9/2 + 4 F 9/2 cross-relaxation between Er 3+ ions (Fig. 2c) and so to a decrease in the overall emission intensity as suggested in Ref. [START_REF] Ju | An upconversion nanoprobe operating in the first biological window[END_REF].

A thermal treatment at (T g +20 • C) for 30 min to 17h was performed in order to check if Ag NPs can grow in these glasses as in Refs. [START_REF] Kuusela | Impact of ZnO addition on Er 3+ nearinfrared emission, the formation of Ag nanoparticles, and the crystallization of sodium fluorophosphate glass[END_REF][START_REF] Ennouri | Investigations of the thermal, structural, and Near-IR emission properties of Ag containing fluorophosphate glasses and their crystallization process[END_REF] during a thermal treatment, such as fiber drawing process for example. The growth of the Ag NPs in the investigated glasses during the thermal treatment was confirmed from the transmittance spectra of the heat treated glasses (Fig. 3). A large absorption band at ~400 nm appears, the intensity of which increases with an increase in x and in the duration of the thermal treatment. This absorption band can be related to the SPR absorption band of Ag NPs [START_REF] Gangopadhyay | Growth of silver nanoclusters embedded in soda glass matrix[END_REF]. During the thermal treatment, the Ag + ions are suspected to change into neutral Ag atoms which aggregate to form Ag NPs. The SPR absorption band shifts to longer wavelength when the duration of the heat treatment increases. These changes in the absorption spectra after the thermal treatment suggest an increase in the number and size of the Ag NPs during the thermal treatment as suggested in Ref. [START_REF] Kuusela | Impact of ZnO addition on Er 3+ nearinfrared emission, the formation of Ag nanoparticles, and the crystallization of sodium fluorophosphate glass[END_REF]. For long thermal treatment, the SPR bands exhibit a long tail revealing the formation of Ag NPs with non-uniform sizes. One should point out the loss in the transmittance of the glasses after the thermal treatment, especially for the Ag4 glass.

The elemental mapping of Ag obtained using SEM is presented in

Table 2

Thermal properties of the preform and the fiber.

T g ( • C) ± 3 • C T x ( • C) ± 3 • C T p ( • C) ± 3 • C ΔT = T x -T g ± 6 • C Ag4 preform 288 383 412 95
Ag4 fiber 280 373 409 93 Fig. 4. It clearly shows that Ag species are well dispersed in the asprepared glass, taken Ag4 as an example (Fig. 4a). After heat treatment, a larger amount of Ag is found at the surface of the glass (Fig. 4b) as in Ref. [START_REF] Ennouri | Investigations of the thermal, structural, and Near-IR emission properties of Ag containing fluorophosphate glasses and their crystallization process[END_REF]. No noticeable changes in the shape nor in the intensity of the upconversion bands were observed although enhancement of the UC emission intensities was reported after the Ag NPs formation in tellurite glasses [START_REF] Yang | Upconversion luminescence properties of Er 3+ -doped TeO 2 -PbF 2 glass with and without Ag nanoparticles[END_REF]. As the glasses were crushed into powder to allow the comparison of the UC intensities, the layer of Ag NPs is suspected to be too thin even after 17h at (T g +20 • C) to impact the upconversion properties of the glasses when measured from powder. The glasses were also heat treated at (T g +20 • C) for 17h and 340 • C for 1h in order to grow CaF 2 crystals as in Ref. [START_REF] Ojha | Tailoring the glass composition to increase the thermal stability without impacting the crystallization behavior of oxyfluorophosphate glass[END_REF] in order to check if the change in the glass composition has an impact on the nucleation and growth mechanism. The XRD pattern of all the heat treated glasses exhibit the peaks related to CaF 2 crystals (Fig. 5). No XRD peaks of Ag could be detected indicating that the amount of Ag nanoparticles in the glasses is too small to be detected.

The volume precipitation of CaF 2 crystals is evidenced in Fig. 6 which shows the SEM and elemental mapping of the cross-section of the Ag0 and Ag4 glasses after heat treatment. Smaller CaF 2 crystals are seen in the heat treated Ag4 glass than in the Ag0 glass.

The precipitation of CaF 2 crystals was also confirmed from the changes in the UC properties of the glasses. The thermal treatment increases the intensity of the upconversion (Fig. 2b). The thermal treatment has also an impact on the shape of the UC spectra. As shown in Fig. 7, the intensity of the red emission is larger than that of the green one after the thermal treatment. These changes in the UC properties of the glasses are again a clear sign of changes in the local symmetry of Er 3+ ions when the glass network changes from amorphous to crystalline. As suggested in Refs. [START_REF] Ojha | Tailoring the glass composition to increase the thermal stability without impacting the crystallization behavior of oxyfluorophosphate glass[END_REF][START_REF] Ojha | Nucleation and growth behavior of Er 3+ doped oxyfluorophosphate glasses[END_REF], the precipitation of Er 3+ doped CaF 2 crystals is suspected to lead to cooperative up-conversion process between Er 3+ ions [START_REF]Rare-Earth-Doped Fiber Lasers and Amplifiers, Revised and Expanded[END_REF]. However, based on the Er mapping (Fig. 6), some Er 3+ ions are expected to remain in the glass matrix. The precipitation of CaF 2 crystals is thus thought to lead to cross-relaxation between Er 3+ ions promoting the red emission of Er 3+ as discussed in the previous paragraph. However, as depicted in Fig. 2b, the increase in the integrated upconversion emission area from 500 to 700 nm is less as x increases. The depolymerization of the glass network and/or the presence of Ag NPs limits the formation of the CaF 2 crystals limiting the increase in the intensity of the UC emissions. Similar results were reported when adding MgO, Al 2 O 3 or Fe 2 O 3 in the same glass network [START_REF] Ojha | Nucleation and growth behavior of Er 3+ doped oxyfluorophosphate glasses[END_REF].

As the addition of Ag 2 O in the oxyfluorophosphate glass network increases the thermal stability of the glass, the Ag4 glass was drawn into single core fibers from preform. Uncoated fibers with large diameter were planned to ease the characterization of the crystals if any were found in the fibers and/or at their surface. As the uncoated fibers were brittle, it was not possible to measure the optical losses. Fig. 8 present the SEM image of the fiber and the composition analysis across the fiber. The drawing process has no noticeable impact on the composition of the glass within the accuracy of the measurement (±1.5 mol %) (Fig. 8b). However, small agglomerates of Ag were found at the surface of the fiber (Fig. 9). One should point out the homogeneous distribution of Na, P, Ca and Er at the surface of the fiber. No crystals were seen in the fiber crosssection (Fig. 8) nor on the surface of the fiber (Fig. 9).

As SEM is not appropriate to quantify F, the thermal and structural properties of the fiber were measured and compared to those of the preform. As shown in Table 2, a slight decrease in T g and T x is observed after drawing. This decrease in T g can be related to the depolymerization of the glass network induced by the drawing process as evidenced by the slight decrease in intensity of the band at 1250 cm -1 compared to the main band (Fig. 10a). Similar changes in the thermal and structural properties of phosphate glasses after drawing were reported in Ref. [START_REF] Gumenyuk | Superluminescence and spectral hole burning effect in ultra-short length Er/Yb-doped phosphate fiber[END_REF]. No sharp peaks can be seen in the IR spectrum of the fiber suggesting that the network remains mainly amorphous during the drawing process as suspected from the SEM analysis.

The transmittance and emission of the as-drawn fibers with three different lengths are presented in Fig. 10b andc, respectively. The transmittance spectrum exhibits the same absorption bands related to Er 3+ ions as in Fig. 1a. The spectra confirm the confinement of the light in the fibers. The emission spectrum of the 1.5 cm long fiber exhibits the typical emission band of Er 3+ corresponding to the transition from the excited 4 I 13/2 level to the 4 I 15/2 level of Er 3+ . However, as the length of the fiber increases, the shape of the emission band changes due to reabsorption of the short wavelengths as reported in Ref. [START_REF] Gumenyuk | Superluminescence and spectral hole burning effect in ultra-short length Er/Yb-doped phosphate fiber[END_REF].

In order to investigate the possibility to prepare glass-ceramic fiber, the fibers were first heat treated for 5 and 30 min at 300 • C, which corresponds to (T g +20 • C), as performed on bulks. However, these heat treatments, performed at lower temperature than the temperatures used for the nucleation and growth of CaF 2 crystals, lead to a significant decrease in the transmittance (Fig. 11) and in the intensity of the emission band (Fig. 12). The degradation in the emission properties of the fiber after heat treatment can be related to an increase in the surface roughness due to the formation of a large amount of Ag NPs at the surface of the fiber as depicted in Fig. 13. As shown in Fig. 14, the heat treatment leads also to an inhomogeneous distribution of the elements in the cross-section of the fiber suggesting that crystallization might have occurred during the thermal treatment despite the low temperature used for the thermal treatment. One should mention that, there is no sign of the formation of CaF 2 crystals in the fiber suggesting that the temperature of the heat treatment is probably too low to induce the precipitation of this CaF 2 crystal.

Conclusion

As a summary, it was demonstrated that Ag 2 O can be used to increase the thermal properties of the Er 3+ doped glass with the composition 75NaPO 3 -25CaF 2 (in mol%) without impacting the volume precipitation of CaF 2 crystals during thermal treatment. Ag nanoparticles were found to precipitate at the surface of the glasses during a thermal treatment performed at temperature near the glass transition temperature.

Although the introduction of Ag 2 O leads to the depolymerization of the phosphate network, the sites of Er 3+ ions remain unchanged. The addition of Ag 2 O in the oxyfluorophosphate network is suspected to limit the precipitation of CaF 2 crystals during thermal treatment. We demonstrate that optical fibers can be drawn from the glass prepared with 4 mol% of Ag 2 O without apparent crystallization nor changes in the glass composition. However, Ag nanoparticles form at the surface of the fiber during the drawing although the drawing is a fast process. We also show that this fiber cannot be thermally treated into glass-ceramic fibers due to the formation of additional Ag nanoparticles at the surface of the fiber increasing the optical losses.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .

 1 Fig. 1. Normalized FTIR spectra of the glasses (a) and absorption spectra of the glasses in the IR (b) and in the UV-Vis (c) range.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Normalized upconversion spectra of the glasses (a) and integrated upconversion emission area from 500 to 700 nm prior to (black) and after heat treatment (T g +20 • C) for 17h and 320 • C for 1h (red) (b) (λ exc = 980 nm). Proposed energy transfer mechanism between Er 3+ ions (c). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Ag elemental mapping measured in the cross-section of Ag4 glass prior to (a) and after (b) thermal treatment at (T g +20 • C) for 17h

Fig. 7 .

 7 Fig. 7. Normalized upconversion spectra of the glasses after heat treatment at (T g +20 • C) for 17h and 340 • C for 1h (λ exc = 980 nm).

Fig. 8 .

 8 Fig. 8. SEM image of the as-drawn fiber (a) and composition analysis measured across the fiber (b).

Fig. 9 .

 9 Fig. 9. SEM image of the fiber surface and mapping of the elements.

Fig. 10 .

 10 Fig. 10. FTIR (a), Transmittance (b) and Emission (c) spectra (λ exc = 980 nm) of the as-drawn fibers with different lengths. FTIR spectrum of the preform is also shown in (a).

Fig. 11 .

 11 Fig. 11. Transmittance spectra of the fiber prior to and after heat treatment with different lengths: 1.5 cm (a), 3 cm (b) and 6 cm (c).

Fig. 12 .

 12 Fig. 12. Emission spectra of the fiber prior to and after heat treatment (λ exc = 980 nm) with different lengths: 1.5 cm (a), 3 cm (b) and 6 cm (c).

Fig. 13 .

 13 Fig. 13. SEM image of the surface of the fiber heat treated for 5 and 30 min at 300 • C and mapping of the elements.

Acknowledgment

Academy of Finland (Flagship Programme, Photonics Research and Innovation PREIN-320165), Pirkanmaa Regional Fund and Magnus Ehrnrooth foundation are greatly acknowledged for the financial support.


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In vivo evaluation of antinociceptive effects of cyriotoxin-1a, the first toxin purified from Cyriopagopus schioedtei spider venom

Over the past two decades, animal venom toxins have been widely explored as an original source of new antinociceptive drugs directed towards the Nav1.7 subtype of voltage-gated sodium channels. This subtype, expressed in afferent sensitive fibers and more particularly in dorsal root ganglia (DRG) neurons, has been validated as an antinociceptive target of choice by human genetic evidence such as congenital insensivity to pain [START_REF] Vetter | Nav1.7 as a pain targetfrom gene to pharmacology[END_REF].

High throughput screening of Smartox biotechnology company collection venoms, using automated patch-clamp (QPatch HTX, Sophion Biosciences) on HEK cells overexpressing human Nav subtypes, pointed out a new inhibitory cysteine knot (ICK) toxin, named cyriotoxin-1a (CyrTx-1a). In DRG neurons isolated from adult mice, the peptide preferentially inhibited with high affinity tetrodotoxin (TTX)-sensitive Na current (IC50 = 170 nM), flowing through mainly the Nav1.7 subtype, compared to TTX-resistant Na current (IC50 =108 µM), flowing through Nav1.8 and Nav1.9 subtypes. In addition, the peptide exhibited nanomolar range affinity for Nav1.7, Nav1.1-1.3 and NaV1.6 subtypes and micromolar range affinity for Nav1.5 and Nav1.4 subtypes. The hot plate and von Frey pain assays showed that mice injected with 367 µg/kg of CyrTx-1a were more resistant to pain than animals injected with the vehicle (PBS).

However, compound muscle action potential (CMAP) recordings after toxin injection revealed a relative narrow therapeutic window w/o side-effects.

In conclusion, the pharmacological profile of CyrTx-1a is of great interest since it leads to further engineering studies aimed to improve its selectivity for optimizing the use of this peptide as an antinociceptive therapeutics.

Ribbon representation of the backbone peptide folding of CyrTx-1a determined by 1 H 2D NMR method. The structural topology of the double-stranded antiparallel β-sheet (arrows) was identified by Pymol. The three disulfide bonds are C2-C17, C9-C22 and C16-C29 (in yellow). Hydrophobic residues are colored in green, and basic and acidic residues are colored in blue and red, respectively. The other polar residues are colored in white.

  Tânia C. Gonçalves, Evelyne Benoit, Rachid Boukaiba, Gerhard Hessler, Denis Servent, Michel Partiseti To cite this version: Tânia C. Gonçalves, Evelyne Benoit, Rachid Boukaiba, Gerhard Hessler, Denis Servent, et al.. In vivo evaluation of antinociceptive effects of cyriotoxin-1a, the first toxin purified from Cyriopagopus schioedtei spider venom. 1st International symposium of Paris-Saclay University -Drug Discovery and New Therapeutics, Apr 2018, Orsay, France. hal-04466080 In vivo evaluation of antinociceptive effects of cyriotoxin-1a, the first toxin purified from Cyriopagopus schioedtei spider venom Tânia C. Gonçalves a,b , Evelyne Benoit b,c , Rachid Boukaiba a , Gerhard Hessler d , Denis Servent b , Michel Partiseti a a Sanofi Aventis R & D, Integrated Drug Discovery -High Content Biology, F-94440 Vitry-sur-Seine, France b Service d'Ingénierie Moléculaire des Protéines (SIMOPRO), CEA, Université Paris-Saclay, F-91191 Gif sur Yvette, France c Institut des Neurosciences Paris-Saclay (Neuro-PSI), UMR CNRS/Université Paris-Sud 9197, Université Paris-Saclay, F-91198 Gif sur Yvette, France d Sanofi Aventis R & D, Integrated Drug Discovery -Synthetic Molecular Design, Frankfurt, Germany