INTEGRAL REPRESENTATION FOR SOME GENERALIZED POLY-CAUCHY NUMBERS

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INTEGRAL REPRESENTATION FOR SOME GENERALIZED POLY-CAUCHY NUMBERS

J Chikhi

To cite this version:

J Chikhi. INTEGRAL REPRESENTATION FOR SOME GENERALIZED POLY-CAUCHY NUM-

BERS. 2016. �hal-01370757�

INTEGRAL REPRESENTATION FOR SOME GENERALIZED POLY-CAUCHY NUMBERS

J. CHIKHI

Abstract. In this note, we establish an integral representation for a special functionE_s,α(z)and apply it to some generalized poly-Cauchy numbersc^(s)_n,α:=n![tⁿ]E_s,α(log(1+t)). We recover, in the special caseα=s=1, the integral representation of the Bernoulli numbers of the second kindbn=c⁽¹⁾_n,1/n! obtained by Feng Qi by quite different methods.

1. INTRODUCTION

Letα andsbe parameters, withα real and postive andscomplex withℜs>0. We define some generalized poly-Cauchy numbers, see [3] fors=kinteger, by the generating function

E_s,α(log(1+t)) =

∞

∑

n=0

c^(s)_n,αtⁿ

n! , (|t|<1), whereEs,α is the function, defined for any complexe numberz, by

Es,α(z):=

∞ n=0

∑

zⁿ n!(n+α)^s .

This function is called poly-exponnential in [1] and, fors=ka positive integer, the extended polylogarithm factorial function, see [3] for instance.

In the sequel, we

•obtain a first integral representation forE_s,α(z)and use it to obtain the main integral representation,

•apply it to get an integral representation for the generalized poly-Cauchy numberc^(s)_n,α,

•recover, by specialisation, the F. Qi’s integral representation for the Cauchy numberc⁽¹⁾_n,1. 2. MELLIN TYPEINTEGRAL REPRESENTATIONS

The firts integral representation is basic and, more or less well known.

Proposition 1. The function E_s,α(z)admits the following integral representation, Es,α(z) = 1

Γ(s) Z 1

0

(−logx)^s−1x^α−1e^zxdx. (1)

Proof. We firts obtain a Mellin type integral representation. Asα andℜsare positive, we have for any non negative integern,

Γ(s) (n+α)^s =

Z ∞

0

t^s−1e^−(n+α)tdt, so

Es,α(z) =

∞ n=0

∑

zⁿ

n!(n+α)^s = 1 Γ(s)

∞ n=0

∑

zⁿ n!

Z _∞

0

t^s−1e^−(n+α)tdt.

Date: September 23, 2016.

2010 Mathematics Subject Classification: 11B83 , 11B34 , 26A48.

Key words and phrases : poly-Cauchy numbers, second kind bernoulli numbers, Mellin transform, integral representation.

1

2 J. CHIKHI

The convergence modes of the series and integral permit to exchangeΣand^R and obtain E_s,α(z) = 1

Γ(s) Z ∞

0

t^s−1e^−αt

∞ n=0

∑

zⁿe^−nt n!

! dt

= 1 Γ(s)

Z ∞

0

t^s−1e^−αte^ze−t dt.

At the end, the changee^−t=x gives the desired formula.

3. MAIN INTEGRAL REPRESENTATION

Here is our main result.

Theorem 2. Let a and b be real numbers and z=a+ib, then

Es,α(z) = e^a 2iπ

Z _∞

0

E_s,α(z₁(u))−E_s,α(z₂(u)) u(u+e^a) du, (2)

where z1(u) =logu+i(b+π)and z2(u) =logu+i(b−π).

Proof. As mentionned bellow, we shall use the first integral representation (1), E_s,α(z) = 1

Γ(s) Z 1

0

(−logx)^s−1x^α−1e^zxdx

= 1 Γ(s)

Z 1 0

(−logx)^s−1x^α−1e^ibxe^axdx.

We putβ =e^a, write

E_s,α(z) = β Γ(s)

Z 1 0

(−logx)^s−1x^α−1e^ibxβ^x−1dx, and use the lemma

Lemma 3.

β^x−1=sin(πx) π

Z ∞

0

u^x−1

u+βdu (0<x<1). (3)

Indeed, we find in many tables of integrals, as [2], the Mellin integral transform expression

Z ∞

0

t^s−1

t+1dt= π

sin(πs) (0<ℜs<1),

and just transform it, by the change of variableu=βt, to get the formula (3) for any real numberx∈(0,1).

Hence, we have

Es,α(z) = β πΓ(s)

Z 1 0

(−logx)^s−1x^α−1e^ibxsin(πx) _{Z ∞}

0

u^x−1 u+βdu

dx, and, by the Fubini’s theorem, that

E_s,α(z) = β πΓ(s)

Z ∞

0

Z 1 0

(−logx)^s−1x^α−1e^ibxu^xsin(πx)dx du

u(u+β)du.

We after consider the integral inside,

Z 1 0

(−logx)^s−1x^α−1e^ibxu^xsin(πx)dx

= 1 2iπ

Z ₁ 0

(−logx)^s−1x^α−1e^ibxu^x(e^iπx−e^−iπx)dx

= 1 2iπ

Z 1 0

(−logx)^s−1x^α−1e(logu+i(b+π))xdx− 1 2iπ

Z 1 0

(−logx)^s−1x^α−1e^{(logu+i(b−π))x}dx

= 1

2iπ(E_s,α(logu+i(b+π))−E_s,α(logu+i(b−π))),

and we are done .

4. THE GENERALIZED POLY-CAUCHY NUMBER INTEGRAL REPRESENTATION

Lett∈(−1,1)and putz=log(1+t). Thenz₁(u) =logu+iπ,z₂(u) =logu−iπand the integral representation (4) writes,

Corollary 4.

E_s,α(log(1+t)) =1+t 2iπ

Z ∞

0

E_s,α(logu+iπ)−E_s,α(logu−iπ)

u(u+1+t) du,

(4)

Remark 1. If s is a real positive number, then E_s,α(log(1+t)) =1+t

π Im Z ∞

0

E_s,α(logu+iπ) u(u+1+t) du, (5)

where Im stands for the imaginary part.

In order to get successive derivatives, int, under the sign^R, that is legitimate, we write that(1+t)(u+1+t)⁻¹= 1−u(u+1+t)⁻¹and obtain for any positive integern,

dⁿ

dtⁿEs,α(log(1+t)) =(−1)ⁿ⁻¹n!

2iπ Z ∞

0

Es,α(logu+iπ)−Es,α(logu−iπ) (u+1+t)ⁿ⁺¹ du. Lettingt=0, we obtain the integral representation for the generalized poly-Cauchy numbers, Corollary 5.

c^(s)_n,α

n! =(−1)ⁿ⁻¹ 2iπ

Z ∞

0

Es,α(logu+iπ)−Es,α(logu−iπ) (u+1)ⁿ⁺¹ du. (6)

5. THEQI INTEGRAL REPRESENTATIONS

For the special caseα=s=1, we have E_1,1(z) =

∞ n=0

∑

zⁿ

(n+1)!=e^z−1 z , then

ImE_1,1(logu+iπ)) =Im −u−1

logu+iπ = π(u+1) log²u+π²

,

and finally by (5),

E_1,1(log(1+t)) = t

log(1+t)= (1+t) Z ∞

0

u+1

u(log²u+π²)(u+1+t)du, (7)

4 J. CHIKHI and by (6),

c⁽¹⁾_n,1

n! = (−1)ⁿ⁻¹ Z ∞

0

du

(log²u+π²)(u+1)ⁿ . (8)

The formulae (7) and (8) above are the integral representations found in [4] by different methods.

6. FINAL REMARK

When the parametersis real, so are the numbersc^(s)_n,α, one could investigate, as done by Feng Qi, the complete mono- tonicity, the log-convexity and, may be, more other proprieties of these numbers by using the integral representations (5) and (6).

REFERENCES

1. K.N. Boyadzhiev,Polyexponentialsavailable online at https://arxiv.org/pdf/0710.1332v1.pdf 2. I.S. Gradshteyn, I.M. Ryzhik,Table of Integrals, Series, and Products, Academic Press.

3. T. Komatsu, V. Laohakosol, K. Liptai,A generalization of poly-Cauchy numbers and their proprieities, Abstract and Applied Analysis, Volume 2013, Article ID 179841, 8 pages.

4. F. Qi, Feng Qi,An integral representation and properties of Bernoulli numbers of the second kind, available online at http://arxiv.org/abs/1301.7181.

JAMELCHIKHI, D ´EPARTEMENT DE MATH_EMATIQUES´ , UNIVERSIT_{E D}´ ’EVRYVAL D’ESSONNE, B ˆATIMENT

I.B.G.B.I. , 23 BD. DEFRANCE, 91037 EVRYCEDEX, FRANCE, E-mail address:jamel.chikhi@univ-evry.fr