The main object of this paper is to investigate a new class of the generalizedq-poly- Bernoulli numbers and polynomials with a parameter

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 355–363 DOI: 10.18514/MMN.2018.2435

SOME RESULTS FOR q-POLY-BERNOULLI POLYNOMIALS WITH A PARAMETER

M. MECHACHA, M. A. BOUTICHE, M. RAHMANI, AND D. BEHLOUL Received 27 October, 2017

Abstract. The main object of this paper is to investigate a new class of the generalizedq-poly- Bernoulli numbers and polynomials with a parameter. We give explicit formulas and a recursive method for the calculation of theq-poly-Bernoulli numbers and polynomials. As a consequence, we derive a method for the calculation of the special values at negative integral points of the Arakawa–Kaneko zeta function also known as generalized Hurwitz zeta function.

2010Mathematics Subject Classification: 11B68; 11B73

Keywords: Arakawa-Kaneko zeta function, Poly-Bernoulli numbers, recurrence relations, Stirl- ing numbers

1. INTRODUCTION

Letqbe an indeterminate with0q < 1. Theq-analogue ofxis defined by ŒxqD1 q^x

1 q

withŒ0qD0and limq!1ŒxqDx. Recently Komatsu in [12] introduced and studied a new family of polynomials, calledq-poly-Bernoulli polynomialsB_n;;q^.k/ .´/with a real parameterwhich are defined by the following generating function:

Fq;.tI´/WD

1 e ^tLik;q

1 e ^t

e ^{t ´}D

1

X

nD0

B_n;;q^.k/ .´/tⁿ

nŠ; (1.1) .n0Ik2ZI¤0/

where Lik;q.´/is theq-polylogarithm function [11] defined by Lik;q.´/D

1

X

nD1

´ⁿ Œn^k_q:

Clearly, we have

qlim!1B_n;;q^.k/ .´/DB_n;^.k/.´/;

c 2018 Miskolc University Press

which is the poly-Bernoulli polynomial with aparameter [7], and

qlim!1Lik;q.´/DLik.´/;

which is the ordinary polylogarithm function, defined by Li_k.´/D

1

X

mD1

´^m

m^k: (1.2)

In addition, when ´D0, Bn;^.k/.0/DBn;^.k/ is the poly-Bernoulli number with a parameter. When ´D0 and D1, B_n;1^.k/.0/DBn^.k/ is the poly-Bernoulli number [1–3,10] defined by

Lik.1 e ^t/

1 e ^t D

1

X

nD0

B_n^.k/tⁿ

nŠ; (1.3)

In this paper, we propose to investigate a new class of the generalized q-poly- Bernoulli numbers and polynomials with a parameter which we call .m; q/-poly- Bernoulli polynomials with a parameter. We establish several properties of these polynomials. The study of.m; q/-poly-Bernoulli polynomials with a parameter yields an interesting algorithm for calculatingBn;m^.k/ .´I; q/. As an application, we derive a recursive method for the calculation of the special values at negative integral points of the Arakawa-Kaneko zeta function.

We first recall some basic definitions and some results [8,16] that will be useful in the rest of the paper. The (signed) Stirling numberss .n; i /of the first kind are the coefficients in the following expansion:

x .x 1/ .x nC1/D

n

X

iD0

s .n; i / xⁱ; n1 and satisfy the recurrence relation given by

s .nC1; i /Ds .n; i 1/ ns .n; i / .1i n/ : (1.4) The Stirling numbers of the second kind, denotedS.n; i /are the coefficients in the expansion

xⁿD

n

X

iD0

S.n; i /x .x 1/ .x iC1/ ; n1:

These numbers count the number of ways to partition a set ofnelements into exactly i nonempty subsets.

The exponential generating functions fors.n; i /andS.n; i /are given by

1

X

nDi

s .n; i / ´ⁿ nŠ D 1

i ŠŒln.1C´/ⁱ

357

and 1

X

nDi

S .n; i / ´ⁿ nŠ D 1

i Š e^´ 1i

;

respectively.

The weighted Stirling numbersS_nⁱ.x/of the second kind are defined by (see [5,6]) S_nⁱ.x/D 1

i Šⁱxⁿ

D 1 i Š

i

X

jD0

. 1/^{i j} i j

!

.xCj /ⁿ;

wheredenotes the forward difference operator. The exponential generating func- tion ofS_n^k.x/is given by

1

X

nDi

S_nⁱ.x/´ⁿ nŠ D 1

i Še^x´ e^´ 1i

(1.5) and weighted Stirling numbersS_nⁱ.x/satisfy the following recurrence relation:

S_nⁱ_C1.x/DS_n^{i 1}.x/C.xCi /S_nⁱ.x/ .1in/:

In particular, we have for nonnegative integerr S_nⁱ.0/DS .n; i / andS_nⁱ.r/D

(nCr iCr

)

r

:

where (n

i )

r

denotes ther-Stirling numbers of the second kind [4].

2. THE.m; q/-POLY-BERNOULLI NUMBERS WITH A PARAMETER In order to computeBn;;q^.k/ WDBn;;q^.k/ .0/, we define.m; q/-poly-Bernoulli numbers B^.k/n;m.; q/with a parameterin terms ofm-Stirling numbers of the second kind by:

B^.k/_n;m.; q/D. /ⁿŒmC1^k_q mŠ

n

X

iD0

.mCi /ŠS_nⁱ.m/

. /ⁱŒmCiC1^k_q; m0 (2.1) withB^.k/_0;m.; q/D1andB_n;0^.k/.; q/DB_n;;q^.k/ .

By direct computation from (2.1), we find B_0;m^.k/.; q/D1;

B_1;m^.k/.; q/D.mC1/

q^mC1 1 q^mC2 1

^k m;

B^.k/_2;m.; q/D.mC1/ .mC2/

q^mC1 1 q^mC3 1

^k

2m²C3mC1

q^mC1 1 q^mC2 1

^k Cm²:

The following Theorem gives us a relation between the .m; q/-poly-Bernoulli numbersBn;m^.k/ .; q/andq-poly-Bernoulli numbersBn^.k/.; q/WDBn;;q^.k/ .

Theorem 1. Form0, we have

B^.k/_n;m.; q/D. /^mŒmC1^k_q mŠ

m

X

iD0

s .m; i /B_n^.k/_Ci.; q/

. /ⁱ : (2.2)

Proof. The explicit formula (2.2) can be derived from a known result in [14, p.

681, Corollary 1] for the Stirling transform upon specializing the initial sequence a0;mD mŠ

. /^mŒmC1^k_q:

The next Theorem contains the exponential generating function for .m; q/-poly- Bernoulli numbers with a parameter.

Theorem 2. The exponential generating function forBn;m^.k/ .; q/is given by

1

X

nD0

B^.k/_n;m.; q/tⁿ

nŠ D. /^mCnŒmC1^k_q

mŠ e^mt

e ^t d

dt m

Fq;.tI´/ : Proof. We have

1

X

nD0

B^.k/_n;m.; q/tⁿ

nŠ D. /^mŒmC1^k_q mŠ

m

X

iD0

s .m; i /

1

X

nD0

B_n^.k/_Ci.; q/

. /ⁱ tⁿ nŠ

D. /^mCnŒmC1^k_q mŠ

m

X

iD0

s .m; i / dⁱ dtⁱ

1 e^tLik;q

1 e^t

:

Since [13]

m

X

iD0

s .m; i / d

dt i

De^mt

e ^t d dt

m

;

we get the desired result.

Next, we propose an algorithm, which is based on a three-term recurrence relation, for calculating the.m; q/ poly-Bernoulli numbersB^.k/n;m.; q/with a parameter.

359

Theorem 3. For every integerk, theB^.k/n;m.; q/satisfies the following three-term recurrence relation:

B^.k/_nC1;m.; q/D.mC1/

q^mC1 1 q^mC2 1

^k

B^.k/_n;mC1.; q/ mB_n;m^.k/ .; q/ (2.3) with the initial sequence given byB^.k/_0;m.; q/D1.

Proof. From.2:2/and.1:4/, we have B^.k/_n;mC1.; q/D. /^mC1ŒmC2^k_q

.mC1/Š

mC1

X

iD0

.s .m; i 1/ ms .m; i //B_n^.k/_Ci.; q/

. /ⁱ :

After some simplifications, we find that B^.k/_n;mC1.; q/D 1

ŒmC1^k_q

. /ŒmC2^k_q mC1

1

. /B^.k/_nC1;m.; q/ mB_n;m^.k/ .; q/

:

This evidently equivalent to.2:3/.

Remark1. If we setD1; kD1andq!1, in.2:3/, we get BnC1;mD.mC1/²

.mC2/ Bn;mC1 mBn;m; (2.4) an algorithm for the classical Bernoulli numbers withB1D¹2. See [15] for the case B1D ¹2 .

3. THE.m; q/-POLY-BERNOULLI POLYNOMIALS WITH A PARAMETER Form0, let us consider the.m; q/-poly-Bernoulli polynomials with a parameter B^.k/n;m.´I; q/as follows:

B_n;m^.k/ .´I; q/D

n

X

iD0

. 1/^{n i} n i

!

B^.k/_i;m.; q/ ´^{n i}: (3.1) It is easy to show that the generating function ofBn;m^.k/ .´I; q/is given by

X

n0

B_n;m^.k/ .´I; q/tⁿ

nŠ De ^´tX

n0

B^.k/_n;m.; q/tⁿ nŠ

D 1

mŠ. /^mCnŒmC1^k_qe^{.m ´/t}

e ^t d dt

m

Fq;.tI´/ : Next, we show an explicit formula aboutB^.k/n;m.´I; q/.

Theorem 4. The following formula holds true

B_n;m^.k/ .´I; q/DŒmC1^k_q mŠ

n

X

iD0

. /^{n i} .mCi /Š ŒmCiC1^k_qS_nⁱ

´ Cm

:

Proof. From (3.1), we have B_n;m^.k/ .´I; q/DŒmC1^k_q

mŠ

n

X

iD0

. 1/^{n i} n i

! _i X

jD0

. /^{i j}.mCj /ŠS_i^j.m/

ŒmCjC1^k_q ´^{n i}

DŒmC1^k_q mŠ

n

X

jD0

. /^{n j}.mCj /Š ŒmCjC1^k_q

n

X

iD0

n i

! S_i^j.m/

´

n i

:

By using the relation

S_nⁱ.xCy/D

n

X

lD0

n l

!

S_lⁱ.x/ y^{n l};

we obtain

B^.k/_n;m.´I; q/DŒmC1^k_q. /ⁿ mŠ

n

X

jD0

.mCj /Š

. /^jŒmCjC1^k_qS_n^j ´

Cm

;

we arrive at the desired result.

Theorem 5. The polynomialsB^.k/n;m.´I; q/satisfy the following three-term recur- rence relation:

B^.k/_nC1;m.´I; q/D.mC1/

q^mC1 1 q^mC2 1

^k

B_n;m^.k/_C1.´I; q/

C.´ m/B_n;m^.k/ .´I; q/ ; (3.2) with the initial sequence given by

B^.k/_0;m.´I; q/D1:

Proof. From (3.1), we get d

d´B_n;m^.k/ .´I; q/D

n

X

iD0

.n i / . 1/^{n i} n i

!

B^.k/_i;m.; q/ ´^{n i 1}

Dn

n

X

iD0

. 1/^{n i} n i

!

B^.k/_i;m.; q/ ´^{n i 1}

n

n

X

iD1

. 1/^{n i} n 1 i 1

!

B^.k/_i;m.; q/ ´^{n i 1}:

361

Then

´ d

d´B^.k/_n;m.´I; q/DnB^.k/_n;m.´I; q/

n

n 1

X

iD0

. 1/^{n i 1} n 1 i

!

B_i^.k/_C1;m.; q/ ´^{n i 1}:

Now, using (2.3), we have

´ d

d´B^.k/_n;m.´I; q/DnB^.k/_n;m.´I; q/Cnm

n 1

X

iD0

n 1 i

!

B^.k/_i;m.; q/ . ´/^{n i 1}

n .mC1/

q^mC1 1 q^mC2 1

^{k n 1} X

iD0

n 1 i

!

B^.k/_i;mC1.; q/ . ´/^{n i 1};

which, after simplification, yields

´B^.k/_{n 1;m}.´I; q/DB_n;m^.k/ .´I; q/ .mC1/

q^mC1 1 q^mC2 1

^k

B^.k/_{n 1;mC1}.´I; q/

CmB^.k/_{n 1;m}.´I; q/ ;

which is obviously equivalent to (3.2) and the proof is complete.

As consequence of Theorem 5, one can deduce a three-term recurrence relation for.m; q/-poly-Bernoulli polynomials with a parameterand negative upper indices B^{. k/}n;m .´I; q/.

Corollary 1. TheBn;m^{. k/}.´I; q/satisfies the following three-term recurrence re- lation:

B_n^{. k/}_C1;m.´I; q/D.mC1/

q^mC2 1 q^mC1 1

^k

B^{. k/}_n;mC1.´I; q/

C.´ m/B^{. k/}_n;m .´I; q/ ; with the initial sequence given by

B^{. k/}_0;m .´I; q/D1:

The next result gives a method for the calculation of the special values at negat- ive integral points of the Arakawa–Kaneko zeta function. Recall that the Arakawa- Kaneko zeta function_k.s; x/, fors2C,x > 0,k2Z, is defined by [9]

k.s; x/D 1 .s/

C1Z

0

t^{s 1}Lik 1 e ^t 1 e ^t e ^xtdt

It is well-known that the special values at negative integral points are given in terms of poly-Bernoulli polynomials

_k. n; x/D. 1/ⁿB_n^.k/.x/ ; n0:

We now present the following algorithm for_k. n; x/. We start with the sequence K0;mD1 as the first row of the matrix Kn;m

n;m0. Each entry is determined recursively by

KnC1;m.k; x/D.mC1/^kC1

.mC2/^k Kn;mC1.k; x/C.x m/Kn;m.k; x/:

Then

_k. n; x/D. 1/ⁿKn;0.k; x/

whereKn;0.k; x/are the first column of the matrix Kn;m

n;m0. 4. CONCLUSION

In our present research, we have investigated a new class of the generalized q- poly-Bernoulli numbers and polynomials with a parameter. As a consequence, we derive a method for the calculation of the special values at negative integral points of the Arakawa-Kaneko zeta function. We have also given a recursive method for the calculation ofq-poly-Bernoulli numbers and polynomials with parameter.

ACKNOWLEDGEMENTS

The authors would like to thank an anonymous referee for the helpful comments.

REFERENCES

[1] T. Arakawa, T. Ibukiyama, and M. Kaneko,Bernoulli numbers and zeta functions. With an ap- pendix by Don Zagier. Tokyo: Springer, 2014. doi:10.1007/978-4-431-54919-2.

[2] A. Bayad and Y. Hamahata, “Polylogarithms and poly-Bernoulli polynomials.”Kyushu J. Math., vol. 65, no. 1, pp. 15–24, 2011, doi:10.2206/kyushujm.65.15.

[3] B. B´enyi and P. Hajnal, “Combinatorics of poly-Bernoulli numbers.”Stud. Sci. Math. Hung., vol. 52, no. 4, pp. 537–558, 2015, doi:10.1556/012.2015.52.4.1325.

[4] A. Z. Broder, “The r-Stirling numbers.” Discrete Math., vol. 49, pp. 241–259, 1984, doi:

10.1016/0012-365X(84)90161-4.

[5] L. Carlitz, “Weighted Stirling numbers of the first and second kind. I.”Fibonacci Q., vol. 18, pp.

147–162, 1980.

[6] L. Carlitz, “Weighted Stirling numbers of the first and second kind. II.”Fibonacci Q., vol. 18, pp.

242–257, 1980.

[7] M. Cenkci and T. Komatsu, “Poly-Bernoulli numbers and polynomials with aqparameter.”J.

Number Theory, vol. 152, pp. 38–54, 2015, doi:10.1016/j.jnt.2014.12.004.

[8] L. Comtet, “Advanced combinatorics. The art of finite and infinite expansions. Translated from the French by J. W. Nienhuys. Rev. and enlarged ed.” Dordrecht, Holland - Boston, U.S.A.: D.

Reidel Publishing Company. X, 343 p. Dfl. 65.00 (1974)., 1974.

[9] M.-A. Coppo and B. Candelpergher, “The Arakawa-Kaneko zeta function.”Ramanujan J., vol. 22, no. 2, pp. 153–162, 2010, doi:10.1007/s11139-009-9205-x.

363

[10] M. Kaneko, “Poly-Bernoulli numbers.”J. Th´eor. Nombres Bordx., vol. 9, no. 1, pp. 221–228, 1997, doi:10.5802/jtnb.197.

[11] M. Katsurada, “Complete asymptotic expansions for certain multiple q-integrals and q- differentials of Thomae-Jackson type.” Acta Arith., vol. 152, no. 2, pp. 109–136, 2012, doi:

10.4064/aa152-2-1.

[12] T. Komatsu, “q-poly-Bernoulli numbers and q-poly-Cauchy numbers with a parameter by Jackson’s integrals.” Indag. Math., New Ser., vol. 27, no. 1, pp. 100–111, 2016, doi:

10.1016/j.indag.2015.08.004.

[13] M. Mohammad-Noori, “Some remarks about the derivation operator and generalized Stirling num- bers.”Ars Comb., vol. 100, pp. 177–192, 2011.

[14] M. Rahmani, “Generalized Stirling transform.”Miskolc Math. Notes, vol. 15, no. 2, pp. 677–690, 2014.

[15] M. Rahmani, “Onp-Bernoulli numbers and polynomials.”J. Number Theory, vol. 157, pp. 350–

366, 2015, doi:10.1016/j.jnt.2015.05.019.

[16] H. M. Srivastava and J. Choi,Zeta andq-zeta functions and associated series and integrals. Am- sterdam: Elsevier, 2012.

Authors’ addresses

M. Mechacha

USTHB, Faculty of Mathematics, BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria E-mail address:mmechacha@usthb.dz

M. A. Boutiche

USTHB, Faculty of Mathematics, BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria E-mail address:mboutiche@usthb.dz

M. Rahmani

USTHB, Faculty of Mathematics, BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria E-mail address:mrahmani@usthb.dz

D. Behloul

USTHB, Department of Computer Science, BP 32, El Alia, 16111 Bab Ezzouar, Algiers, Algeria E-mail address:behloul.djilali@gmail.com