Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials

DOI10.1016/j.camwa.2005.04.018zbMath1099.33011OpenAlexW2044270276MaRDI QIDQ2507027

Qiu-Ming Luo, Hari M. Srivastava

Publication date: 10 October 2006

Published in: Computers \& Mathematics with Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.camwa.2005.04.018

zbMATH Keywords

generating functionsgeneralized Bernoulli numbersBernoulli numbersStirling numbers of the second kindEuler numbersBernoulli polynomialsgeneralized Bernoulli polynomialsApostol-Bernoulli polynomialsgeneralized Apostol-Bernoulli polynomialsgeneralized Apostol-Euler polynomialsgeneralized Euler polynomialsEuler polynomialsApostol-Euler polynomialsgeneralized Euler numbersApostol-Bernoulli numbersApostol-Euler numbersgeneralized Apostol-Bernoulli numbersgeneralized Apostol-Euler numbersSrivastava-Pintér addition theorems

Mathematics Subject Classification ID

Bell and Stirling numbers (11B73) Bernoulli and Euler numbers and polynomials (11B68) Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45)

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Cites Work

• Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials
• A note on the Bernoulli and Euler polynomials
• Remarks on some relationships between the Bernoulli and Euler polynomials.
• On the Apostol-Bernoulli polynomials
• The special functions and their approximations. Vol. I, II
• On the Lerch zeta function
• A Class of Addition Theorems^†
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

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