Multiplication formulas for q-Appell polynomials and the multiple q-power sums

Thomas Ernst


In the first article on q-analogues of two Appell polynomials, the generalized Apostol-Bernoulli  and Apostol-Euler  polynomials, focus was on generalizations, symmetries, and complementary argument theorems. In this second article, we focus on a recent paper by Luo, and one paper on power sums by Wang and Wang. Most of the proofs are made by using generating functions, and the (multiple) q-addition plays a fundamental role. The introduction of the q-rational numbers in formulas with q-additions enables natural q-extension of vector forms of Raabes multiplication formulas. As special cases, new formulas for q-Bernoulli and q-Euler polynomials are obtained.


Raabes multiplication formulas; q-Appell polynomials; multiple q-power sum; symmetry; q-rational number

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Apostol, T. M., On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167.

Carlitz, L., A note on the multiplication formulas for the Bernoulli and Euler polynomials, Proc. Amer. Math. Soc. 4 (1953), 184-188.

Ernst. T., A Comprehensive Treatment of q-calculus, Birkhauser/Springer, Basel, 2012.

Ernst, T., On certain generalized q-Appell polynomial expansions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 68, No. 2 (2015), 27-50.

Ernst, T., A solid foundation for q-Appell polynomials, Adv. Dyn. Syst. Appl. 10 (2015), 27-35.

Ernst, T., Expansion formulas for Apostol type q-Appell polynomials, and their special cases, submitted.

Luo, Q.-M., Srivastava, H. M., Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl. 308, No. 1 (2005), 290-302.

Luo, Q-M., Srivastava, H. M., Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51, No. 3-4 (2006), 631-642.

Luo, Q.-M., Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10, No. 4 (2006), 917-925.

Luo, Q.-M., The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Integral Transforms Spec. Funct. 20, No. 5-6 (2009), 377-391.

Milne-Thomson, L. M., The Calculus of Finite Differences, Macmillan and Co., Ltd., London, 1951.

Wang, Weiping, Wang, Wenwen, Some results on power sums and Apostol-type polynomials, Integral Transforms Spec. Funct. 21, No. 3-4 (2010), 307-318.

Data publikacji: 2016-07-04 15:43:59
Data złożenia artykułu: 2016-06-30 20:36:50


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