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)