Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Bernoulli and Euler Polynomials in Two Variables

  • Received : 2022.08.11
  • Accepted : 2022.11.08
  • Published : 2024.03.31
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Abstract

In a previous work we studied generalized Stirling numbers of the second kind S(a2,b2,p2)a1,b1 (p1, k), where a1, a2, b1, b2 are given complex numbers, a1, a2 ≠ 0, and p1, p2 are non-negative integers given. In this work we use these generalized Stirling numbers to define Bernoulli polynomials in two variables Bp1,p2 (x1, x2), and Euler polynomials in two variables Ep1p2 (x1, x2). By using results for S(1,x2,p2)1,x1 (p1, k), we obtain generalizations, to the bivariate case, of some well-known properties from the standard case, as addition formulas, difference equations and sums of powers. We obtain some identities for bivariate Bernoulli and Euler polynomials, and some generalizations, to the bivariate case, of several known identities for Bernoulli and Euler numbers and polynomials of the standard case.

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Acknowledgement

I thank the anonymous referee for his/her careful reading of the original version of this work. The many details and observations he/she pointed out, certainly contributed to present this final version.

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