Duran, Ugur

Sadjang, P. Njionou

English

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 20 (2019), No. 2, pp. 1185–1199 DOI: 10.18514/MMN.2019.2587ON TWO BIVARIATE KINDS OF .p;q/-BERNOULLIPOLYNOMIALSP. NJIONOU SADJANG AND UGUR DURANReceived 08 April, 2018Abstract. The main aim of this paper is to introduce and investigate .p;q/-extensions of twobivariate kinds of Bernoulli polynomials and numbers. We firstly examine several .p;q/-analoguesof the Taylor expansions of products of some trigonometric functions and determine their coef-ficients which are also analyzed in detail. Then, we introduce two bivariate kinds of .p;q/-Bernoulli polynomials and acquired multifarious formulas and relations including connectionformulas, recurrence formulas, correlations with aforementioned coefficients, partial .p;q/-differentialequations and .p;q/-integral representations.2010 Mathematics Subject Classification: 33D15; 33D45; 11B68; 11B73; 11B83Keywords: trigonometric functions, Bernoulli numbers and polynomials, .p;q/-calculus, Cauchyproduct, series manipulation procedure1. INTRODUCTIONThe Appell polynomials An.x/ defined byf .t/ext D1XnD0An.x/tnnŠ; (1.1)where f is a formal power series in t , have found remarkable applications in differentbranches of mathematics, theoretical physics and chemistry, see [1, 3–13] and refer-ences cited therein. One of the most famous polynomials of the of Appell families isBernoulli polynomials Bn.x/, generated by f .t/D tet  1 in (1.1). Also, Bernoullinumbers, denoted by Bn WD Bn.0/ are of considerable importance in number theory,combinatorics and numerical analysis (cf. [1, 3, 5, 6, 8–10]). Further, they have thefollowing exponential generating function:tet  1 D1XnD0BntnnŠ.jt j< 2/;c 2019 Miskolc University Press1186 P. NJIONOU SADJANG AND UGUR DURANand can be generated by the following recurrence relationnXkD0 nC1k!Bk D 0 for n 1 and B0 D 1:Bernoulli numbers are directly related to several combinatorial numbers such as Stirl-ing, Cauchy and harmonic numbers. For example, except B1 we haveBn D . 1/nnXmD0. 1/mmŠmC1 S2.n;m/; (1.2)whereS2.n;m/D 1mŠmXjD0. 1/j mj!.m j /n;denote the second kind of Stirling numbers [9] with S2.n;m/D 0 for n < m. Veryrecently in [7], Jamei et al. introduced a new kind of bivariate Bernoulli polynomialsand studied their main properties. As a valuable application of these extended poly-nomials, they introduced an extension of the well-known Euler-Maclaurin quadratureformula. Rahmani [8] defined a new family of p-Bernoulli numbers, which are de-rived from the Gaussian hypergeometric function, and established some basic prop-erties. Based on a three-term recurrence relation, he gave an algorithm for computingBernoulli numbers and presented a similar algorithm for Bernoulli polynomials.The Bernoulli polynomials and numbers have found diverse extensions such aspoly-Bernoulli numbers, which are somehow connected to multiple zeta values. Theq-extension of Bernoulli numbers and polynomials has now found many applicationsin combinatorics, statistics and various branches of applied mathematics. Mahmudov[5, 6] introduced a class of generalized Bernoulli polynomials, Euler polynomialsand Genocchi polynomials based on the q-integers and obtained the q-analogues ofwell-known formulas including q-analogue of the Srivastava-Pinter addition theoremand correlations with the q-Bernstein polynomials. Duran et al. [3] considered thenew generating functions of the Bernoulli, the Euler and the Genocchi polynomialsunder post quantum calculus, denoted by .p;q/-calculus. From those generatingfunctions, they analyzed their various behaviours and derived a relation between thenew and old polynomials by making use of the fermionic p-adic integral over thep-adic number fields. Njionou [11] introduced the .p;q/-Appell polynomials whichcovers generalizations of some famous family of polynomials such as the .p;q/-Bernoulli, the .p;q/-Euler and the .p;q/-Genocchi polynomials. He provided severalcharacterizations of these polynomials.In this paper, we introduce a .p;q/-extension of the aforesaid bivariate Bernoullipolynomials and establish their properties. Multifarious connections and inversionformulas are stated and proved. In the following section, some preliminaries andON TWO BIVARIATE KINDS OF .p;q/-BERNOULLI POLYNOMIALS 1187definitions are given and in Section 3, a bivariate kind of .p;q/-Bernoulli polynomi-als is introduced and some of its fundamental properties are stated and proved.2. PRELIMINARIES AND DEFINITIONSLet N denotes the set of all natural numbers, R denotes the set of all real numbersandC denotes the set of all complex numbers. Let us introduce the following notation(see [4, 10, 12])Œnp;q D pn qnp q ; 0 < jqj< jpj  1for any positive integer.The twin-basic number is a natural generalization of the q-number, that islimp!1Œnp;q D Œnq:The .p;q/-factorial is defined byŒnp;qŠDnYkD1Œkp;qŠ; n 1; Œ0p;qŠD 1:Let us introduce also the so-called .p;q/-binomial coefficientsnkp;qD Œnp;qŠŒkp;qŠŒn kp;qŠ ; 0 k  n:Note that as p! 1, the .p;q/-binomial coefficients reduce to the q-binomial coeffi-cients.It is clear by definition that nkp;qDnn kp;q:Let us introduce also the so-called falling and raising .p;q/-powers respectively [4,12].x	a/np;q D .x a/.px aq/   .xpn 1 aqn 1/;.x˚a/np;q D .xCa/.pxCaq/   .xpn 1Caqn 1/:These definitions are extended to.a	b/1p;q D1YkD0.apk  qkb/;.a˚b/1p;q D1YkD0.apkCqkb/;where the convergence is required.1188 P. NJIONOU SADJANG AND UGUR DURANDefinition 1 ([4,12]). Let f be an arbitrary function and a be a real number, thenthe .p;q/-integral of f is defined byZ a0f .x/dp;qx D .p q/a1XkD0qkpkC1f qkpkC1a!if 0 < jqj< jpj  1:Let f be a function defined on the set of the complex numbers.Definition 2 ([4, 12]). The .p;q/-derivative of the function f is defined asDp;qf .x/D f .px/ f .qx/.p q/x ; x ¤ 0;and .Dp;qf /.0/D f 0.0/, provided that f is differentiable at 0.Proposition 1. The .p;q/-derivative operator fulfills the following product andquotient rulesDp;q.f .x/g.x//D f .px/Dp;qg.x/Cg.qx/Dp;qf .x/;Dp;q.f .x/g.x//D g.px/Dp;qf .x/Cf .qx/Dp;qg.x/:Dp;qf .x/g.x/D g.qx/Dp;qf .x/ f .qx/Dp;qg.x/g.px/g.qx/;Dp;qf .x/g.x/D g.px/Dp;qf .x/ f .px/Dp;qg.x/g.px/g.qx/:Proposition 2. If F.x/ is a .p;q/-antiderivative of f .x/ and F.x/ is continuousat x D 0, we haveZ baf .x/dp;qx D F.b/ F.a/; 0 a < b 1:Corollary 1. If f 0.x/ exists in a neighbourhood of x D 0 and is continuous atx D 0, where f 0.x/ denotes the ordinary derivative of f .x/, we haveZ baDp;qf .x/dp;qx D f .b/ f .a/:Proposition 3. Suppose that f .x/ and g.x/ are two functions whose ordinaryderivatives exist in a neighbourhood of x D 0. a and b are two real numbers suchthat a < b , thenZ baf .px/ Dp;qg.x/dp;qxDf .b/g.b/ f .a/g.a/ Z bag.qx/ Dp;qf .x/dp;qx:(2.1)ON TWO BIVARIATE KINDS OF .p;q/-BERNOULLI POLYNOMIALS 1189As in the q-case, there are many definitions of the .p;q/-exponential function. Thefollowing two .p;q/-analogues of the exponential function (see [3, 4, 10, 11]) will befrequently used throughout this paper:ep;q.´/D1XnD0p.n2/Œnp;qŠ´n; (2.2)Ep;q.´/D1XnD0q.n2/Œnp;qŠ´n: (2.3)From the definitions (2.2) and (2.3) of the .p;q/-exponential functions, it is easy tosee that [10]ep;q.x/Ep;q. x/D 1: (2.4)From (2.2) we can deriveep;q.i´/D1XnD0p.n2/Œnp;qŠ.i´/n D1XnD0. 1/np.2n2 /Œ2np;qŠ´2nC i1XnD0. 1/np.2nC12 /Œ2nC1p;qŠ ´2nC1:(2.5)By (2.5), the .p;q/-cosine and the .p;q/-sine functions are defined (see [4, 10]) asfollows:cosp;q.´/D1XnD0. 1/np.2n2 /Œ2np;qŠ´2n; (2.6)sinp;q.´/D1XnD0. 1/np.2nC12 /Œ2nC1p;qŠ ´2nC1: (2.7)In the following definition, we generalize the notion of q-addition introduced byJackson and studied later by Ward and Al-Salam, see [1, 2] for more details. Our.p;q/-addition reduces to the q-addition defined by Euler and recalled in [13]Definition 3. Let x and y be two complex numbers.(1) The .p;q/-addition of x and y which we denote by x p˚;q y is defined by x p˚;q yn D nXkD0nkp;qpk.k n/xkyn k : (2.8)(2) The .p;q/-subtration of x and y which we denote by x	p;q y is defined by x	p;q yn D nXkD0nkp;q. 1/n kpk.k n/xkyn k : (2.9)1190 P. NJIONOU SADJANG AND UGUR DURANTheorem 1. The following relation holds true for any x;y 2 R.ep;q.x/ep;q.y/D ep;q.x p˚;q y/ (2.10)ep;q.x/ep;q. y/D ep;q.x	p;q y/: (2.11)Proof. By the definition of the .p;q/-addition and the Cauchy product we canreadily see thatep;q.x/ep;q.y/D1XkD0p.k2/xkŒkp;qŠ1X`D0p.`2/y`Œ`p;qŠD1XnD0 nXkD0p.k2/C.n k2 /xkyn kŒkp;qŠŒn kp;qŠ!D1XnD0 nXkD0nkp;qpk.k n/xkyn k!p.n2/Œnp;qŠD ep;q.x p˚;q y/:The second assertion is proved in the same way. 3. A BIVARIATE KIND OF .p;q/-BERNOULLI POLYNOMIALSLet x;y 2 R. It is well-known that the Taylor expansion of the two functionsext cosyt and ext sinyt are as follows [7]ext cosyt D1XnD0Cn.x;y/tnnŠ; (3.1)andext sinyt D1XnD0Sn.x;y/tnnŠ; (3.2)whereCn.x;y/Dbn2cXkD0. 1/k n2k!xn 2ky2k; (3.3)andSn.x;y/Dbn 12cXkD0. 1/k n2kC1!xn 2k 1y2kC1: (3.4)Here we introduce a .p;q/-extension of the two above polynomials Cn.x;y/ andSn.x;y/ by the following generating functions:ep;q.xt/cosp;q.yt/D1XkD0Ck;p;q.x;y/tkŒkp;qŠ; (3.5)ON TWO BIVARIATE KINDS OF .p;q/-BERNOULLI POLYNOMIALS 1191andep;q.xt/sinp;q.yt/D1XkD0Sk;p;q.x;y/tkŒkp;qŠ; (3.6)Some particular cases areC2n;p;q.0;y/D . 1/np.2n2 /y2n; C2nC1;p;q.0;y/D 0andS2n;p;q.0;y/D 0; S2nC1;p;q.0;y/D . 1/np.2nC12 /y2nC1:The following Lemma will be useful in the derivation of several results.Lemma 1 ([9]). The following elementary series manipulations hold1XnD01XkD0A.k;n/D1XnD0bn=2cXkD0A.k;n 2k/; (3.7)1XnD01XkD0B.k;n/D1XnD0b.n 1/=2cXkD0B.k;n 1 2k/: (3.8)Theorem 2. The following representations holdCn;p;q.x;y/D p.n2/bn2cXkD0. 1/kn2kp;qp2k.k n/xn 2ky2k (3.9)andSn;p;q.x;y/D p.n 12 /bn 12cXkD0. 1/kn2kC1p;qp4k2 2knxn 2k 1y2kC1 (3.10)Proof. By series manipulation procedure (3.7), we haveep;q.xt/cosp;q.yt/D 1XnD0p.n2/Œnp;qŠ.xt/n! 1XnD0. 1/np.2n2 /Œ2np;qŠ.yt/2n!D1XnD0bn=2cXkD0p.n 2k2 /Œn 2kp;qŠ .xt/n 2k . 1/kp.2k2 /Œ2kp;qŠ.yt/2kD1XnD00@p.n2/ bn2 cXkD0. 1/kn2kp;qp2k.k n/xn 2ky2k1A tnŒnp;qŠ;which proves (3.5). The proof of (3.6) is similar by means of the series manipulationmethod (3.8).1192 P. NJIONOU SADJANG AND UGUR DURANTheorem 3. The following derivative rules are validDp;q;xCk;p;q.x;y/D Œkp;qCk 1;p;q.px;y/; (3.11)Dp;q;yCk;p;q.x;y/D Œkp;qSk 1;p;q.x;py/; (3.12)Dp;q;xSk;p;q.x;y/D Œkp;qSk 1;p;q.px;y/; (3.13)Dp;q;ySk;p;q.x;y/D Œkp;qCk 1;p;q.x;py/: (3.14)Proof. Relation (3.5) yields1XnD1Dp;q;xCn;p;q.x;y/tnŒnp;qŠD tep;q.pxt/cosp;q.yt/D1XnD0Cn;p;q.px;y/tnC1Œnp;qŠD1XnD1Cn 1;p;q.px;y/tnŒn 1p;qŠD1XnD0Œnp;qCn 1;p;q.px;y/tnŒnp;qŠ;proving (3.11). The other equations (3.12), (3.13) and (3.14) can be similarly proved.Theorem 4. The following relations are validCn;p;q.x;y/DnXkD0nkp;qp.n k2 /Ck;p;q.0;y/xn k; (3.15)Sn;p;q.x;y/DnXkD0nkp;qp.n k2 /Sk;p;q.0;y/xn k : (3.16)Theorem 5. The following power representations hold. 1/np.2n2 /y2n D2nXkD0. 1/kq.k2/2nkp;qC2n k;p;q.x;y/xk; (3.17)and. 1/np.2nC12 /y2nC1 D2nC1XkD0. 1/kq.k2/2nC1kp;qS2nC1 k;p;q.x;y/xk :(3.18)Proof. Multiplying both sides of (3.5) by Ep;q. xt/ and using (2.4), it followsthat1XnD0. 1/np.2n2 /y2n t2nŒnp;qŠD 1XnD0q.n2/. x/ntnŒnp;qŠ! 1XnD0Cn;p;q.x;y/tnŒnp;qŠ!ON TWO BIVARIATE KINDS OF .p;q/-BERNOULLI POLYNOMIALS 1193D1XnD0 nXkD0. 1/kq.k2/nkp;qCn k;p;q.x;y/xk!tnŒnp;qŠ;which proves (3.17). The proof of (3.18) is similar. Theorem 6. The following connection formulas holdC2nC1;p;q.x;y/D2nXkD0. 1/kq.kC12 /2nC1kC1p;qC2n k;p;q.x;y/xkC1; (3.19)andS2n;p;q.x;y/D2n 1XkD0. 1/kq.kC12 /2nkC1p;qS2n k 1;p;q.x;y/xkC1: (3.20)Proof. From the relation1XnD0. 1/np.2n2 /y2n t2nŒnp;qŠD1XnD0 nXkD0. 1/kq.k2/nkp;qCn k;p;q.x;y/xk!tnŒnp;qŠ;it follows that2nC1XkD0. 1/kq.k2/2nC1kp;qC2nC1 k;p;q.x;y/xk D 0:Hence (3.19) is proved. We prove (3.20) in the same way.We can now introduce two kinds of bivariate q-Bernoulli polynomials astep;q.xt/ep;q.t/ 1 cosp;q.yt/D1XnD0B.c/n;p;q.x;y/tnŒnp;qŠ; (3.21)andtep;q.xt/ep;q.t/ 1 sinp;q.yt/D1XnD0B.s/n;p;q.x;y/tnŒnp;qŠ: (3.22)Upon setting xDyD 0 for both polynomials in (3.21) and (3.22), we haveB.c/n;p;q.0;0/DB.s/n;p;q.0;0/ WD Bn;p;q which are called .p;q/-Bernoulli polynomials defined in [3].When y D 0 in (3.21) and (3.22), we get the usual .p;q/-Bernoulli polynomials,denoted by Bn;p;q.x/, see [3, 11].Next, we give some basic properties of these polynomials.1194 P. NJIONOU SADJANG AND UGUR DURANTheorem 7. B.c/n;p;q.x;y/ and B.s/n;p;q.x;y/ can be represented in terms of .p;q/-Bernoulli numbers as followsB.c/n;p;q.x;y/DnXkD0nkp;qBk;p;qCn k;p;q.x;y/; (3.23)andB.s/n;p;q.x;y/DnXkD0nkp;qBk;p;qSn k;p;q.x;y/; (3.24)Proof. Using the Cauchy product rule, we have1XnD0B.c/n;p;q.x;y/tnŒnp;qŠD tep;q.t/ 1ep;q.xt/cosp;q.yt/D 1XnD0BntnŒnp;qŠ! 1XnD0Cn;p;q.x;y/tnŒnp;qŠ!D1XnD0 nXkD0nkp;qBk;p;qCn k;p;q.x;y/!tnŒnp;qŠ;which proves (3.23). The proof of (3.24) is similar. We now state the following theorem.Theorem 8. The following connection formulas are validB.c/n;p;q.x;y/DŒn2 XkD0. 1/kn2kp;qBn 2k;p;q.x/p.2k2 /y2k; (3.25)andB.s/n;p;q.x;y/DŒn2 XkD0. 1/kn2kC1p;qBn 1 2k;p;q.x/p.2kC12 /y2kC1: (3.26)Proof. The formula (3.25) follows from (3.7) since1XnD0B.c/n;p;q.x;y/tnŒnqD tep;q.xt/ep;q.t/ 1 cosp;q.yt/D 1XnD0Bn;p;q.x/tnŒnp;qŠ! 1XnD0. 1/np.2n2 /Œ2np;qŠ.yt/2n!D1XnD00@ Œn2 XkD0. 1/kn2kp;qBn 2k;p;q.x/p.2k2 /y2k1A tnŒnp;qŠ:ON TWO BIVARIATE KINDS OF .p;q/-BERNOULLI POLYNOMIALS 1195The proof of (3.26) is similar via (3.8). Theorem 9. The following connection formulas are validCn;p;q.x;y/DnXkD0p.kC12 /ŒkC1p;qnkp;qB.c/n;p;q.x;y/; (3.27)Sn;p;q.x;y/DnXkD0p.kC12 /ŒkC1p;qnkp;qB.s/n;p;q.x;y/: (3.28)Proof. From (3.21), we have1XnD0B.c/n;p;q.x;y/tnŒnp;qŠD tep;q.xt/ep;q.t/ 1 cosp;q.yt/Dtep;q.t/ 11XnD0Cn;p;q.x;y/tnŒnp;qŠ:Hence1XnD0Cn;p;q.x;y/tnŒnp;qŠD ep;q.t/ 1t1XnD0B.c/n;p;q.x;y/tnŒnp;qŠD 1XnD0p.nC12 /ŒnC1p;qtnŒnp;qŠ! 1XnD0B.c/n;p;q.x;y/tnŒnp;qŠ!D1XnD0 nXkD0p.kC12 /ŒkC1p;qnkp;qB.c/n;p;q.x;y/!tnŒnp;qŠ:Thus (3.27) follows. (3.28) is proved in a similar way. Proposition 4. For every n 2N, the following identities holdB.c/n;p;q..1 p˚;q x/;y/ B.c/n;p;q.x;y/D Œnp;qCn 1;p;q.x;y/; (3.29)B.s/n;p;q..1 p˚;q x/;y/ B.s/n;p;q.x;y/D Œnp;qSn 1;p;q.x;y/: (3.30)Proof. We have1XnD0B.c/n;p;q..1 p˚;q x/;y/tnŒnp;qŠD tep;qŒ.1 p˚;q x/t ep;q.t/ 1 cosp;q.yt/D tep;q.xt/Œep;q.t/ 1C1ep;q.t/ 1 cosp;q.yt/D tep;q.xt/cosp;q.yt/C tep;q.xt/ep;q.t/ 1 cosp;q.yt/D1XnD0Cn;p;q.x;y/tnC1Œnp;qŠC1XnD0B.c/n;p;q.x;y/tnŒnp;qŠ;which proves (3.29). The identity (3.30) is proved similarly. 1196 P. NJIONOU SADJANG AND UGUR DURANCorollary 2. The following relations holdB.c/2nC1;p;q.1;y/ B.c/2nC1;p;q.0;y/D Œ2nC1p;q. 1/np.2n2 /y2n;B.s/2n;p;q.1;y/ B.s/2n;p;q.0;y/D Œ2np;q. 1/nC1p.2n 12 /y2n 1:Proof. If we replace n by 2nC1 in (3.29), and x by 0, we obtainB.c/2nC1;p;q.1;y/ B.c/2nC1;p;q.0;y/D Œ2nC1p;qC2n;p;q.0;y/:The first relation is proved since from (3.9) we have C2n;q.0;y/D . 1/np.2n2 /y2n.The second relation is proved similarly. Proposition 5. For every n 2N, the following identities holdB.c/n;p;q..x p˚;q ´/;y/DnXkD0nkp;qBk;p;q.x/Cn k;p;q.y;´/; (3.31)andB.s/n;p;q..x p˚;q ´/;y/DnXkD0nkp;qBk;p;q.x/Sn k;p;q.y;´/; (3.32)Proof. We have1XnD0B.c/n;p;q..x p˚;q ´/;y/tnŒnp;qŠD tep;q..x p˚;q ´/t/ep;q.t/ 1 cosp;q.yt/D tep;q.xt/ep;q.t/ 1  ep;q.´t/cosp;q.yt/D 1XnD0Bn;p;q.x/tnŒnp;qŠ! 1XnD0Cn;p;q.y;´/tnŒnp;qŠ!D1XnD0 nXkD0nkp;qBk;p;q.x/Cn k.y;´/!tnŒnqŠ;which proves (3.31). The proof of (3.32) is similar. Proposition 6. For every n 2N, the following identities holdB.c/n;p;q..x p˚;q ´/;y/DnXkD0nkp;qp.n k2 /B.c/k;p;q.x;y/´n k; (3.33)andB.s/n;p;q..x p˚;q ´/;y/DnXkD0nkp;qp.n k2 /B.s/k;p;q.x;y/´n k : (3.34)ON TWO BIVARIATE KINDS OF .p;q/-BERNOULLI POLYNOMIALS 1197Proof. We have1XnD0B.c/n;p;q..x p˚;q ´/;y/tnŒnp;qŠD tep;q..x p˚;q ´/t/ep;q.t/ 1 cosp;q.yt/D tep;q.xt/ep;q.t/ 1 cosp;q.yt/ ep;q.´t/D 1XnD0B.c/n;p;q.x;y/tnŒnp;qŠ! 1XnD0p.n2/´ntnŒnp;qŠ!D1XnD0 nXkD0nkp;qp.n k2 /B.c/k;p;q.x;y/´n k!tnŒnqŠ;which proves (3.33). The proof of (3.34) is similar. Proposition 7. The following equations can be concludednXkD0nC1kp;qp.nC1 k2 /B.c/k;p;q.x;y/D ŒnC1p;qCn;p;q.x;y/; (3.35)nXkD0nC1kp;qp.nC1 k2 /B.s/k;p;q.x;y/D ŒnC1p;qSn;p;q.x;y/: (3.36)Proof. From (3.33), we haveB.c/nC1;p;q..x p˚;q 1/;y/ B.c/nC1;p;q.x;y/DnXkD0nC1kp;qp.nC1 k2 /B.c/k;p;q.x;y/:Hence, by using (3.29), relation (3.35) is derived. The proof of (3.36) is concludedin a similar way. Corollary 3. Relations (3.35) and (3.36) imply thatnXkD0nC1kp;qp.nC1 k2 /B.c/k;p;q.0;y/D(. 1/mŒ2mC1p;qp.2m2 /y2m if nD 2m is odd;0 if nD 2mC1 is even;andnXkD0nC1kp;qp.nC1 k2 /B.s/k;p;q.0;y/D(0 if nD 2m is odd;. 1/mŒ2mC2p;qp.2mC12 /y2mC1 if nD 2mC1 is even:1198 P. NJIONOU SADJANG AND UGUR DURANCorollary 4. For every n 2N, the following partial .p;q/-differential equationsholdDp;q;xB.c/n;p;q.x;y/D Œnp;qB.c/n 1;p;q.px;y/;Dp;q;yB.c/n;p;q.x;y/D Œnp;qB.c/n 1;p;q.x;py/;Dp;q;xB.s/n;p;q.x;y/D Œnp;qB.s/n 1;p;q.px;y/;andDp;q;yB.c/n;q.x;y/D Œnp;qB.s/n 1;p;q.x;py/: (3.37)Corollary 5. The following equations are validZ 10B.c/2n;p;q.px;y/dp;qx D . 1/np.2n2 /y2n;Z 10B.s/2nC1;p;q.px;y/dp;qx D . 1/nC1p.2nC12 /y2nC1;which are proved by combining Proposition 4 and Corollary 2 using the definition ofthe .p;q/-integral.REFERENCES[1] W. A. Al-Salam, “q-Bernoulli numbers and polynomials,” Mathematicshe Nachrichten, vol. 17,pp. 239–260, 1958.[2] K. S. Chung, W.-S. Chung, S.-T. Nam, and H. J. Kang, “New q-derivative and q-logarithm,”International Journal of Theoretical Physics, vol. 33, no. 10, pp. 2019–2029, 1993.[3] U. Duran, M. Acikgoz, and S. Araci, “On some polynomials derived from .p;q/-calculus,”Journal of Computational and Theoretical Nanoscience, vol. 13, no. 11, pp. 7903–7908, 2016.[4] U. Duran, M. Acikgoz, and S. Araci, “A study on some new results arising from .p;q/-calculus,”TWMS Journal of Pure and Applied Mathematics, 2018.[5] N. I. Mahmudov, “q-Analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pinter addition theorems,” Discrete Dynamics in Nature and Society, 2012.[6] N. I. Mahmudov, “On a class of q-Bernoulli and q-Euler polynomials,” Advance in DifferenceEquations, vol. 108, 2013.[7] M. Masjed-Jamei, M. R. Beyki, and W. Koepf, “On a bivariate kind of Bernoulli polynomaials,”Submitted.[8] M. Rahmani, “On p-Bernoulli numbers and polynomials,” Journal of Number Theory, vol. 157,pp. 350–366, 2017.[9] E. D. Rainville, Special Functions. New York: The Macmillan Company, 1960.[10] M. N. Sadjang, “On two .p;q/-analogues of the Laplace transform,” Journal of Difference Equa-tions and Applications, vol. 23, pp. 1562–1583, 2017.[11] M. N. Sadjang, “On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor theorems,”Results in Mathematics, vol. 73, no. 39, 2018.[12] M. N. Sadjang, “On (p,q)-Appell polynomials,” Analysis Mathematica, in press.[13] M. Schork, “Ward’s ”Calculus of Sequences”, q-calculus and the limit q! 1,” Advanced Studiesin Contemporary Mathematics, vol. 13, no. 2, pp. 131–141, 2006.ON TWO BIVARIATE KINDS OF .p;q/-BERNOULLI POLYNOMIALS 1199Authors’ addressesP. Njionou SadjangP. Njionou Sadjang, Faculty of Industrial Engineering, University of Douala, Douala, CameroonE-mail address: pnjionou@yahoo.frUgur DuranUgur Duran, Department of Basic Science of Engineering, Faculty of Engineereing and, NaturalSciences, Iskenderun Technical University, TR-31200, Hatay, TurkeyE-mail address: mtdrnugur@gmail.com

On two bivariate kinds of (p, q)-Bernoulli polynomials

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