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Discrete Dynamics in Nature and Society
VolumeΒ 2012Β (2012), Article IDΒ 169348, 8 pages
http://dx.doi.org/10.1155/2012/169348
Research Article

π‘ž-Analogues of the Bernoulli and Genocchi Polynomials and the Srivastava-PintΓ©r Addition Theorems

Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey

Received 24 April 2012; Revised 5 July 2012; Accepted 23 July 2012

Academic Editor: Lee ChaeΒ Jang

Copyright Β© 2012 N. I. Mahmudov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli and Genocchi polynomials based on the π‘ž-integers. The π‘ž-analogues of well-known formulas are derived. The π‘ž-analogue of the Srivastava-PintΓ©r addition theorem is obtained.

1. Introduction

Throughout this paper, we always make use of the following notation: β„• denotes the set of natural numbers, β„•0 denotes the set of nonnegative integers, ℝ denotes the set of real numbers, and β„‚ denotes the set of complex numbers.

The π‘ž-shifted factorial is defined by (π‘Ž;π‘ž)0=1,(π‘Ž;π‘ž)𝑛=π‘›βˆ’1𝑗=0ξ€·1βˆ’π‘žπ‘—π‘Žξ€Έ,π‘›βˆˆβ„•,(π‘Ž;π‘ž)∞=βˆžξ‘π‘—=0ξ€·1βˆ’π‘žπ‘—π‘Žξ€Έ,||π‘ž||<1,π‘Žβˆˆβ„‚.(1.1) The π‘ž-numbers and π‘ž-numbers factorial is defined by [π‘Ž]π‘ž=1βˆ’π‘žπ‘Ž[0]1βˆ’π‘ž(π‘žβ‰ 1);π‘ž[𝑛]!=1;π‘ž[1]!=π‘ž[2]π‘žβ‹―[𝑛]π‘žπ‘›βˆˆβ„•,π‘Žβˆˆβ„‚,(1.2) respectively. The π‘ž-polynomial coefficient is defined by βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž=(π‘ž;π‘ž)𝑛(π‘ž;π‘ž)π‘›βˆ’π‘˜(π‘ž;π‘ž)π‘˜.(1.3) The π‘ž-analogue of the function (π‘₯+𝑦)𝑛 is defined by (π‘₯+𝑦)π‘›π‘žβˆΆ=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ‘ž(1/2)π‘˜(π‘˜βˆ’1)π‘₯π‘›βˆ’π‘˜π‘¦π‘˜,π‘›βˆˆβ„•0.(1.4) In the standard approach to the π‘ž-calculus two exponential function are used: π‘’π‘ž(𝑧)=βˆžξ“π‘›=0𝑧𝑛[𝑛]π‘ž!=βˆžξ‘π‘˜=01ξ€·1βˆ’(1βˆ’π‘ž)π‘žπ‘˜π‘§ξ€Έ||π‘ž||1,0<<1,|𝑧|<||||,𝐸1βˆ’π‘žπ‘ž(𝑧)=βˆžξ“π‘›=0π‘ž(1/2)𝑛(π‘›βˆ’1)𝑧𝑛[𝑛]π‘ž!=βˆžξ‘π‘˜=0ξ€·1+(1βˆ’π‘ž)π‘žπ‘˜π‘§ξ€Έ||π‘ž||,0<<1,π‘§βˆˆβ„‚.(1.5) From this form we easily see that π‘’π‘ž(𝑧)πΈπ‘ž(βˆ’π‘§)=1. Moreover, π·π‘žπ‘’π‘ž(𝑧)=π‘’π‘ž(𝑧),π·π‘žπΈπ‘ž(𝑧)=πΈπ‘ž(π‘žπ‘§),(1.6) where π·π‘ž is defined by π·π‘žπ‘“(𝑧)∢=𝑓(π‘žπ‘§)βˆ’π‘“(𝑧).π‘žπ‘§βˆ’π‘§(1.7) The previous π‘ž-standard notation can be found in [1].

Carlitz has introduced the π‘ž-Bernoulli numbers and polynomials in [2]. Srivastava and PintΓ©r proved some relations and theorems between the Bernoulli polynomials and Euler polynomials in [3]. They also gave some generalizations of these polynomials. In [4–6], Kim et al. investigated some properties of the π‘ž-Euler polynomials and Genocchi polynomials. They gave some recurrence relations. In [7], Cenkci et al. gave the π‘ž-extension of Genocchi numbers in a different manner. In [5], Kim gave a new concept for the π‘ž-Genocchi numbers and polynomials. In [8], Simsek et al. investigated the π‘ž-Genocchi zeta function and 𝑙-function by using generating functions and Mellin transformation. We also recall the definitions of the π‘ž-Bernoulli and the π‘ž-Genocchi polynomials of higher order (see [2, 9–12]): (βˆ’π‘‘)π›Όβˆžξ“π‘›=0ξ€·[𝛼]π‘žξ€Έπ‘›[𝑛]π‘ž!π‘žπ‘›+π‘₯𝑒𝑑[𝑛+π‘₯]π‘ž=βˆžξ“π‘›=0𝐡(𝛼)𝑛,π‘ž(𝑑π‘₯)𝑛,𝑛!(2𝑑)π›Όβˆžξ“π‘›=0ξ€·[𝛼]π‘žξ€Έπ‘›[𝑛]π‘ž!(βˆ’1)π‘›π‘žπ‘›+π‘₯𝑒𝑑[𝑛+π‘₯]π‘ž=βˆžξ“π‘›=0𝐺(𝛼)𝑛,π‘žπ‘‘(π‘₯)𝑛.𝑛!(1.8) We propose the following definitions. We define the π‘ž-Bernoulli and the π‘ž-Genocchi polynomials of higher order in two variables π‘₯ and 𝑦, using two π‘ž-exponential functions, which helps us easily prove some properties of these polynomials and π‘ž-analogue of the Srivastava and PintΓ©r addition theorem.

Definition 1.1. The π‘ž-Bernoulli numbers 𝔅(𝛼)𝑛,π‘ž and polynomials 𝔅(𝛼)𝑛,π‘ž(π‘₯,𝑦) in π‘₯,𝑦 of order 𝛼 are defined by means of the generating function functions: ξ‚΅π‘‘π‘’π‘žξ‚Ά(𝑑)βˆ’1𝛼=βˆžξ“π‘›=0𝔅(𝛼)𝑛,π‘žπ‘‘π‘›[𝑛]π‘ž!𝑑,|𝑑|<2πœ‹,π‘’π‘žξ‚Ά(𝑑)βˆ’1π›Όπ‘’π‘ž(𝑑π‘₯)πΈπ‘ž(𝑑𝑦)=βˆžξ“π‘›=0𝔅(𝛼)𝑛,π‘žπ‘‘(π‘₯,𝑦)𝑛[𝑛]π‘ž!,|𝑑|<2πœ‹.(1.9)

Definition 1.2. The π‘ž-Genocchi numbers π”Š(𝛼)𝑛,π‘ž and polynomials π”Š(𝛼)𝑛,π‘ž(π‘₯,𝑦) in π‘₯,𝑦 are defined by means of the generating functions: ξ‚΅2π‘‘π‘’π‘žξ‚Ά(𝑑)+1𝛼=βˆžξ“π‘›=0π”Š(𝛼)𝑛,π‘žπ‘‘π‘›[𝑛]π‘ž!ξ‚΅,|𝑑|<πœ‹,2π‘‘π‘’π‘žξ‚Ά(𝑑)+1π›Όπ‘’π‘ž(𝑑π‘₯)πΈπ‘ž(𝑑𝑦)=βˆžξ“π‘›=0π”Š(𝛼)𝑛,π‘žπ‘‘(π‘₯,𝑦)𝑛[𝑛]π‘ž!,|𝑑|<πœ‹.(1.10)

It is obvious that 𝔅(𝛼)𝑛,π‘ž=𝔅(𝛼)𝑛,π‘ž(0,0),limπ‘žβ†’1βˆ’π”…(𝛼)𝑛,π‘ž(π‘₯,𝑦)=𝐡𝑛(𝛼)(π‘₯+𝑦),limπ‘žβ†’1βˆ’π”…(𝛼)𝑛,π‘ž=𝐡𝑛(𝛼),π”Š(𝛼)𝑛,π‘ž=π”Š(𝛼)𝑛,π‘ž(0,0),limπ‘žβ†’1βˆ’π”Š(𝛼)𝑛,π‘ž(π‘₯,𝑦)=𝐺𝑛(𝛼)(π‘₯+𝑦),limπ‘žβ†’1βˆ’π”Š(𝛼)𝑛,π‘ž=𝐺𝑛(𝛼).(1.11) Here 𝐡𝑛(𝛼)(π‘₯) and 𝐸𝑛(𝛼)(π‘₯) denote the classical Bernoulli, and Genocchi polynomials of order 𝛼 are defined by ξ‚€π‘‘π‘’π‘‘ξ‚βˆ’1𝛼𝑒𝑑π‘₯=βˆžξ“π‘›=0𝐡𝑛(𝛼)(𝑑π‘₯)𝑛,ξ‚€2𝑛!𝑒𝑑+1𝛼𝑒𝑑π‘₯=βˆžξ“π‘›=0𝐺𝑛(𝛼)(𝑑π‘₯)𝑛.𝑛!(1.12)

The aim of the present paper is to obtain some results for the π‘ž-Genocchi polynomials (properties of the π‘ž-Bernoulli polynomials are studied in [13]). The π‘ž-analogues of well-known results, for example, Srivastava and PintΓ©r [3], can be derived from these π‘ž-identities. It should be mentioned that probabilistic proofs the Srivastava-PintΓ©r addition theorems were given recently in [14]. The formulas involving the π‘ž-Stirling numbers of the second kind, π‘ž-Bernoulli polynomials and π‘ž-Bernstein polynomials, are also given. Furthermore some special cases are also considered.

The following elementary properties of the π‘ž-Genocchi polynomials π”ˆ(𝛼)𝑛,π‘ž(π‘₯,𝑦) of order 𝛼 are readily derived from Definition 1.2. We choose to omit the details involved.

Property 1.3. Special values of the π‘ž-Genocchi polynomials of order 𝛼: π”ˆ(0)𝑛,π‘ž(π‘₯,0)=π‘₯𝑛,π”ˆ(0)𝑛,π‘ž(0,𝑦)=π‘ž(1/2)𝑛(π‘›βˆ’1)𝑦𝑛.(1.13)

Property 1.4. Summation formulas for the π‘ž-Genocchi polynomials of order 𝛼: π”ˆ(𝛼)𝑛,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ”ˆ(𝛼)π‘˜,π‘ž(π‘₯+𝑦)π‘žπ‘›βˆ’π‘˜,π”ˆ(𝛼)𝑛,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ”ˆ(π›Όβˆ’1)π‘›βˆ’π‘˜,π‘žπ”ˆπ‘˜,π‘žπ”Š(π‘₯,𝑦),(𝛼)𝑛,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ‘ž(π‘›βˆ’π‘˜)(π‘›βˆ’π‘˜βˆ’1)/2π”Š(𝛼)π‘˜,π‘ž(π‘₯,0)π‘¦π‘›βˆ’π‘˜=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ”Š(𝛼)π‘˜,π‘ž(0,𝑦)π‘₯π‘›βˆ’π‘˜,π”Š(𝛼)𝑛,π‘ž(π‘₯,0)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ”Š(𝛼)π‘˜,π‘žπ‘₯π‘›βˆ’π‘˜,π”Š(𝛼)𝑛,π‘ž(0,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ‘ž(π‘›βˆ’π‘˜)(π‘›βˆ’π‘˜βˆ’1)/2π”Š(𝛼)π‘˜,π‘žπ‘¦π‘›βˆ’π‘˜.(1.14)

Property 1.5. Difference equations: π”Š(𝛼)𝑛,π‘ž(1,𝑦)+π”Š(𝛼)𝑛,π‘ž[𝑛](0,𝑦)=2π‘žπ”Š(π›Όβˆ’1)π‘›βˆ’1,π‘žπ”Š(0,𝑦),(𝛼)𝑛,π‘ž(π‘₯,0)+π”Š(𝛼)𝑛,π‘ž[𝑛](π‘₯,βˆ’1)=2π‘žπ”Š(π›Όβˆ’1)π‘›βˆ’1,π‘ž(π‘₯,βˆ’1).(1.15)

Property 1.6. Differential relations: π·π‘ž,π‘₯π”Š(𝛼)𝑛,π‘ž[𝑛](π‘₯,𝑦)=π‘žπ”Š(𝛼)π‘›βˆ’1,π‘ž(π‘₯,𝑦),π·π‘ž,π‘¦π”Š(𝛼)𝑛,π‘ž[𝑛](π‘₯,𝑦)=π‘žπ”Š(𝛼)π‘›βˆ’1,π‘ž(π‘₯,π‘žπ‘¦).(1.16)

Property 1.7. Addition theorem of the argument: π”ˆ(𝛼+𝛽)𝑛,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žπ”ˆ(𝛼)π‘›βˆ’π‘˜,π‘ž(π‘₯,0)π”ˆ(𝛽)π‘˜,π‘ž(0,𝑦).(1.17)

Property 1.8. Recurrence relationships: π”Š(𝛼)𝑛,π‘žξ‚€1π‘šξ‚+,π‘¦π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘›βˆ’π‘˜π”Š(𝛼)π‘˜,π‘ž[𝑛](0,𝑦)=2π‘žπ‘›βˆ’1ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘›βˆ’1π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘›βˆ’1βˆ’π‘˜π”Š(π›Όβˆ’1)π‘˜,π‘ž(0,𝑦).(1.18)

2. Explicit Relationship between the π‘ž-Genocchi and the π‘ž-Bernoulli Polynomials

In this section we prove an interesting relationship between the π‘ž-Genocchi polynomials π”Š(𝛼)𝑛,π‘ž(π‘₯,𝑦) of order 𝛼 and the π‘ž-Bernoulli polynomials. Here some π‘ž-analogues of known results will be given. We also obtain new formulas and their some special cases in the following.

Theorem 2.1. For π‘›βˆˆβ„•0, the following relationship π”Š(𝛼)𝑛,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=01π‘šπ‘›βˆ’π‘˜βˆ’1[]π‘˜+1π‘žβŽ‘βŽ’βŽ’βŽ’βŽ£2[]π‘˜+1π‘žπ‘˜ξ“π‘—=0βŽ‘βŽ’βŽ’βŽ£π‘˜π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘˜βˆ’π‘—π”Š(π›Όβˆ’1)𝑗,π‘žβˆ’(π‘₯,βˆ’1)π‘˜+1𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘˜+1π‘ž1π‘šπ‘˜+1βˆ’π‘—π”Š(𝛼)𝑗,π‘ž(π‘₯,βˆ’1)βˆ’π”Š(𝛼)π‘˜+1,π‘žβŽ€βŽ₯βŽ₯βŽ₯βŽ¦π”…(π‘₯,0)π‘›βˆ’π‘˜,π‘ž(0,π‘šπ‘¦)(2.1) holds true between the π‘ž-Genocchi and the π‘ž-Bernoulli polynomials.

Proof. Using the following identity: ξ‚΅2π‘‘π‘’π‘žξ‚Ά(𝑑)+1π›Όπ‘’π‘ž(𝑑π‘₯)πΈπ‘žξ‚΅(𝑑𝑦)=2π‘‘π‘’π‘žξ‚Ά(𝑑)+1π›Όπ‘’π‘žπ‘’(𝑑π‘₯)β‹…π‘ž(𝑑/π‘š)βˆ’1π‘‘β‹…π‘‘π‘’π‘ž(𝑑/π‘š)βˆ’1β‹…πΈπ‘žξ‚€π‘‘π‘šξ‚,π‘šπ‘¦(2.2) we have βˆžξ“π‘›=0π”Š(𝛼)𝑛,π‘žπ‘‘(π‘₯,𝑦)𝑛[𝑛]π‘ž!=π‘šπ‘‘βˆžξ“π‘›=0βŽ›βŽœβŽœβŽœβŽπ‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘›βˆ’π‘˜π”Š(𝛼)π‘˜,π‘ž(π‘₯,0)βˆ’π”Š(𝛼)𝑛,π‘žβŽžβŽŸβŽŸβŽŸβŽ π‘‘(π‘₯,0)𝑛[𝑛]π‘ž!βˆžξ“π‘›=0𝔅𝑛,π‘žπ‘‘(0,π‘šπ‘¦)π‘›π‘šπ‘›[𝑛]π‘ž!=βˆžξ“π‘›=1βŽ›βŽœβŽœβŽœβŽπ‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘›βˆ’1βˆ’π‘˜π”Š(𝛼)π‘˜,π‘ž(π‘₯,0)βˆ’π‘šπ”Š(𝛼)𝑛,π‘žβŽžβŽŸβŽŸβŽŸβŽ π‘‘(π‘₯,0)π‘›βˆ’1[𝑛]π‘ž!βˆžξ“π‘›=0𝔅𝑛,π‘žπ‘‘(0,π‘šπ‘¦)π‘›π‘šπ‘›[𝑛]π‘ž!=βˆžξ“π‘›=0βŽ›βŽœβŽœβŽœβŽπ‘›+1ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘›+1π‘žπ‘šπ‘˜π”Š(𝛼)π‘˜,π‘ž(π‘₯,0)βˆ’π‘šπ‘›+1π”Š(𝛼)𝑛+1,π‘žβŽžβŽŸβŽŸβŽŸβŽ π‘‘(π‘₯,0)π‘›π‘šπ‘›[]𝑛+1π‘ž!βˆžξ“π‘›=0𝔅𝑛,π‘žπ‘‘(0,π‘šπ‘¦)π‘›π‘šπ‘›[𝑛]π‘ž!=βˆžξ“π‘›π‘›=0ξ“π‘˜=01π‘šπ‘›[]π‘˜+1π‘žβŽ›βŽœβŽœβŽœβŽπ‘˜+1𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘˜+1π‘žπ‘šπ‘—π”Š(𝛼)𝑗,π‘ž(π‘₯,0)βˆ’π‘šπ‘˜+1π”Š(𝛼)π‘˜+1,π‘žβŽžβŽŸβŽŸβŽŸβŽ π”…(π‘₯,0)π‘›βˆ’π‘˜,π‘žπ‘‘(0,π‘šπ‘¦)𝑛[𝑛]π‘ž!.(2.3) It remains to use Property 1.8.

Since π”Š(𝛼)𝑛,π‘ž(π‘₯,𝑦) is not symmetric with respect to π‘₯ and 𝑦, we can prove a different form of the previously mentioned theorem. It should be stressed out that Theorems 2.1 and 2.2 coincide in the limiting case when π‘žβ†’1βˆ’.

Theorem 2.2. For π‘›βˆˆβ„•0, the following relationship π”Š(𝛼)𝑛,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘›βˆ’π‘˜βˆ’1[]π‘˜+1π‘žβŽ‘βŽ’βŽ’βŽ’βŽ£2[]π‘˜+1π‘žπ‘˜ξ“π‘—=0βŽ‘βŽ’βŽ’βŽ£π‘˜π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘˜βˆ’π‘—π”Š(π›Όβˆ’1)𝑗,π‘žβˆ’(0,𝑦)π‘˜+1𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘˜+1π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘˜+1βˆ’π‘—π”Š(𝛼)𝑗,π‘ž(0,𝑦)βˆ’π”Šπ‘˜+1,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦(0,𝑦)Γ—π”…π‘›βˆ’π‘˜,π‘ž(π‘šπ‘₯,0)(2.4) holds true between the π‘ž-Genocchi and the π‘ž-Bernoulli polynomials.

Proof. The proof is based on the following identity: ξ‚΅2π‘‘π‘’π‘žξ‚Ά(𝑑)+1π›Όπ‘’π‘ž(𝑑π‘₯)πΈπ‘žξ‚΅(𝑑𝑦)=2π‘‘π‘’π‘žξ‚Ά(𝑑)+1π›ΌπΈπ‘žπ‘’(𝑑𝑦)β‹…π‘ž(𝑑/π‘š)βˆ’1π‘‘β‹…π‘‘π‘’π‘ž(𝑑/π‘š)βˆ’1β‹…π‘’π‘žξ‚€π‘‘π‘šξ‚.π‘šπ‘₯(2.5)

Next we discuss some special cases of Theorems 2.1 and 2.2. By noting that π”Š(0)𝑗,π‘ž(0,𝑦)=π‘ž(1/2)𝑗(π‘—βˆ’1)𝑦𝑗,π”Š(0)𝑗,π‘ž(π‘₯,βˆ’1)=(π‘₯βˆ’1)π‘—π‘ž,(2.6) we deduce from Theorems 2.1 and 2.2 Corollary 2.3 below.

Corollary 2.3. For π‘›βˆˆβ„•0, π‘šβˆˆβ„• the following relationship π”Šπ‘›,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘›βˆ’π‘˜βˆ’1[]π‘˜+1π‘žβŽ‘βŽ’βŽ’βŽ’βŽ£2[]π‘˜+1π‘žπ‘˜ξ“π‘—=0βŽ‘βŽ’βŽ’βŽ£π‘˜π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘˜βˆ’π‘—π‘ž(1/2)𝑗(π‘—βˆ’1)π‘¦π‘—βˆ’π‘˜+1𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘˜+1π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘˜+1βˆ’π‘—π”Šπ‘—,π‘ž(0,𝑦)βˆ’π”Šπ‘˜+1,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦(0,𝑦)Γ—π”…π‘›βˆ’π‘˜,π‘žπ”Š(π‘šπ‘₯,0),𝑛,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘›βˆ’π‘˜βˆ’1[]π‘˜+1π‘žβŽ‘βŽ’βŽ’βŽ’βŽ£2[]π‘˜+1π‘žπ‘˜ξ“π‘—=0βŽ‘βŽ’βŽ’βŽ£π‘˜π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘˜βˆ’π‘—(π‘₯βˆ’1)π‘—π‘žβˆ’π‘˜+1𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘˜+1π‘ž1π‘šπ‘˜+1βˆ’π‘—π”Šπ‘—,π‘ž(π‘₯,βˆ’1)βˆ’π”Šπ‘˜+1,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦(π‘₯,0)Γ—π”…π‘›βˆ’π‘˜,π‘ž(0,π‘šπ‘¦)(2.7) holds true between the π‘ž-Bernoulli polynomials and π‘ž-Euler polynomials.

Corollary 2.4. For π‘›βˆˆβ„•0, π‘šβˆˆβ„• the following relationship holds true: 𝐺𝑛(π‘₯+𝑦)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ 2ξ€·π‘˜+1(π‘˜+1)π‘¦π‘˜βˆ’πΊπ‘˜+1,π‘žξ€Έπ΅(𝑦)π‘›βˆ’π‘˜(π‘₯),(2.8)𝐺𝑛(π‘₯+𝑦)=π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ 1π‘šπ‘›βˆ’π‘˜βˆ’1(π‘˜+1)2(π‘˜+1)πΊπ‘˜ξ‚€1𝑦+π‘šξ‚βˆ’1βˆ’πΊπ‘˜+1ξ‚€1𝑦+π‘šξ‚βˆ’1βˆ’πΊπ‘˜+1𝐡(𝑦)π‘›βˆ’π‘˜,π‘ž(π‘šπ‘₯)(2.9) between the classical Genocchi polynomials and the classical Bernoulli polynomials.

Note that the formula (2.9) is new for the classical polynomials.

In terms of the π‘ž-Genocchi numbers π”Š(𝛼)π‘˜,π‘ž, by setting 𝑦=0 in Theorem 2.1, we obtain the following explicit relationship between the π‘ž-Genocchi polynomials π”Š(𝛼)π‘˜,π‘ž of order 𝛼 and the π‘ž-Bernoulli polynomials.

Corollary 2.5. The following relationship holds true: π”Š(𝛼)𝑛,π‘ž(π‘₯,0)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž1π‘šπ‘›βˆ’π‘˜βˆ’1[]π‘˜+1π‘žβŽ‘βŽ’βŽ’βŽ’βŽ£2[]π‘˜+1π‘žπ‘˜ξ“π‘—=0βŽ‘βŽ’βŽ’βŽ£π‘˜π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘˜βˆ’π‘—π”Š(π›Όβˆ’1)𝑗,π‘žβˆ’π‘˜+1𝑗=0βŽ‘βŽ’βŽ’βŽ£π‘—βŽ€βŽ₯βŽ₯βŽ¦π‘˜+1π‘žξ‚€1π‘šξ‚βˆ’1π‘žπ‘˜+1βˆ’π‘—π”Š(𝛼)𝑗,π‘žβˆ’π”Š(𝛼)π‘˜+1,π‘žβŽ€βŽ₯βŽ₯βŽ₯βŽ¦π”…π‘›βˆ’π‘˜,π‘ž(π‘šπ‘₯,0).(2.10)

Corollary 2.6. For π‘›βˆˆβ„•0 the following relationship holds true: π”Šπ‘›,π‘ž(π‘₯,𝑦)=π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž2[]π‘˜+1π‘žξ€Ί[]π‘˜+1π‘žπ‘ž(1/2)π‘˜(π‘˜βˆ’1)π‘¦π‘˜βˆ’π”Šπ‘˜+1,π‘žξ€»π”…(0,𝑦)π‘›βˆ’π‘˜,π‘ž(π‘₯,0).(2.11)

Corollary 2.7. For π‘›βˆˆβ„•0 the following relationship holds true: π”Šπ‘›,π‘ž(π‘₯,0)=βˆ’π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž2[]π‘˜+1π‘žπ”Šπ‘˜+1,π‘žπ”…π‘›βˆ’π‘˜,π‘žπ”Š(π‘₯,0),𝑛,π‘ž=βˆ’π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£π‘›π‘˜βŽ€βŽ₯βŽ₯βŽ¦π‘ž2[]π‘˜+1π‘žπ”Šπ‘˜+1,π‘žπ”…π‘›βˆ’π‘˜,π‘ž.(2.12)

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