Bernoulli and Euler Polynomials in Two Variables
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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 Ep1,p2(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.
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 Ep1,p2(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.
참고문헌 (13)
- L. Carlitz, "Problem 795", Math. Mag., 44, pp.107~ (1971)
- G. -S. Cheon, "A Note on the Bernoulli and Euler polynomials", Appl. Math. Lett., 16(3), pp.365~368 (2003)
- L. Comtet, "Advanced Combinatorics", pp.354~ (1974), D. Reidel Publishing Co.
- K. Dilcher, "General convolution identities for Bernoulli and Euler polynomials", J. Math. Anal. Appl., 435(2), pp.1478~1498 (2016)
- H. W. Gould, "Explicit formulas for Bernoulli numbers", Amer. Math. Monthly, 79, pp.44~51 (1972)
- B. N. Guo, "Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind", J. Comput. Appl. Math., 272, pp.251~257 (2014)
- B. C. Kellner, "Identities between polynomials related to Stirling and harmonic numbers", Integers, 14, pp.22~ (2014)
- T. Komatsu, "Several formulas for Bernoulli numbers and polynomials", Adv. Math. Commun., 17(3), pp.522~535 (2023)
- "NIST Digital Library of Mathematical Functions" (2021)[URL]
- C. Pita-Ruiz, "On a family of Euler type numbers and polynomials", Rend. Mat. Appl. (7), 44, pp.1~26 (2023)
- C. Pita-Ruiz, "On bi-variate poly-Bernoulli polynomials", Commun. Math., 31(1), pp.179~203 (2023)
- J. Worpitzky, "Studienuber die Bernoullischen und Eulerschen Zahlen", J. Reine Angew. Math., 94, pp.203~232 (1883)