On certain generalized q-Appell polynomial expansions

Thomas Ernst

Annales UMCS, Mathematica (2015)

  • Volume: 68, Issue: 2, page 27-50
  • ISSN: 2083-7402

Abstract

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We study q-analogues of three Appell polynomials, the H-polynomials, the Apostol-Bernoulli and Apostol-Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials.

How to cite

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Thomas Ernst. "On certain generalized q-Appell polynomial expansions." Annales UMCS, Mathematica 68.2 (2015): 27-50. <http://eudml.org/doc/270016>.

@article{ThomasErnst2015,
abstract = {We study q-analogues of three Appell polynomials, the H-polynomials, the Apostol-Bernoulli and Apostol-Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials.},
author = {Thomas Ernst},
journal = {Annales UMCS, Mathematica},
keywords = {q-Apostol-Bernoulli polynomials; q-Apostol-Euler polynomials; q-H-polynomials; complementary argument theorem; generating function; multiple q-Appell polynomials; -Apostol-Bernoulli polynomials; -Apostol-Euler polynomials; -H-polynomials; multiple -Appell polynomials},
language = {eng},
number = {2},
pages = {27-50},
title = {On certain generalized q-Appell polynomial expansions},
url = {http://eudml.org/doc/270016},
volume = {68},
year = {2015},
}

TY - JOUR
AU - Thomas Ernst
TI - On certain generalized q-Appell polynomial expansions
JO - Annales UMCS, Mathematica
PY - 2015
VL - 68
IS - 2
SP - 27
EP - 50
AB - We study q-analogues of three Appell polynomials, the H-polynomials, the Apostol-Bernoulli and Apostol-Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials.
LA - eng
KW - q-Apostol-Bernoulli polynomials; q-Apostol-Euler polynomials; q-H-polynomials; complementary argument theorem; generating function; multiple q-Appell polynomials; -Apostol-Bernoulli polynomials; -Apostol-Euler polynomials; -H-polynomials; multiple -Appell polynomials
UR - http://eudml.org/doc/270016
ER -

References

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  2. [2] Dere, R., Simsek, Y., Srivastava, H. M., A unified presentation of three families of generalized Apostol type polynomials based upon the theory of the umbral calculus and the umbral algebra, J. Number Theory 133, no. 10 (2013), 3245-3263.[WoS] Zbl1295.11023
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  4. [4] Ernst, T., q-Pascal and q-Wronskian matrices with implications to q-Appell polynomials, J. Discrete Math. 2013. Zbl1295.05060
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  9. [9] 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. Zbl1099.33011
  10. [10] Luo, Q.-M., Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10, no. 4 (2006), 917-925. Zbl1189.11011
  11. [11] Milne-Thomson, L. M., The Calculus of Finite Differences, Macmillan and Co., Ltd., London, 1951. Zbl59.1111.01
  12. [12] Nørlund, N. E., Differenzenrechnung, Springer-Verlag, Berlin, 1924. 
  13. [13] Pintér, Á, Srivastava, H. M., Addition theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math. 85, no. 3 (2013), 483-495.[WoS] Zbl1288.11022
  14. [14] Sandor, J., Crstici, B., Handbook of number theory II, Kluwer Academic Publishers, Dordrecht, 2004. Zbl1079.11001
  15. [15] Srivastava, H. M., Özarslan, M. A., Kaanoglu, C., Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Russ. J. Math. Phys. 20, no. 1 (2013), 110-120.[WoS] Zbl1292.11043
  16. [16] Wang, W., Wang, W., Some results on power sums and Apostol-type polynomials, Integral Transforms Spec. Funct. 21, no. 3-4 (2010), 307-318. Zbl1203.33011
  17. [17] Ward, M., A calculus of sequences, Amer. J. Math. 58 (1936), 255-266. Zbl62.0408.03

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