On certain generalized q-Appell polynomial expansions

Thomas Ernst

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2014)

  • Volume: 68, Issue: 2
  • ISSN: 0365-1029

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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 68.2 (2014): null. <http://eudml.org/doc/289766>.

@article{ThomasErnst2014,
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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
language = {eng},
number = {2},
pages = {null},
title = {On certain generalized q-Appell polynomial expansions},
url = {http://eudml.org/doc/289766},
volume = {68},
year = {2014},
}

TY - JOUR
AU - Thomas Ernst
TI - On certain generalized q-Appell polynomial expansions
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2014
VL - 68
IS - 2
SP - null
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
UR - http://eudml.org/doc/289766
ER -

References

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  1. Apostol, T. M., On the Lerch zeta function, Pacific J. Math. 1 (1951), 161–167. 
  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. 
  3. Ernst, T., A comprehensive treatment of q-calculus, Birkhäuser, Basel, 2012. 
  4. Ernst, T., q-Pascal and q-Wronskian matrices with implications to q-Appell polynomials, J. Discrete Math. 2013. 
  5. Jordan, Ch., Calculus of finite differences, Third Edition, Chelsea Publishing Co., New York, 1950. 
  6. Kim M., Hu S., A note on the Apostol–Bernoulli and Apostol–Euler polynomials, Publ. Math. Debrecen 5587 (2013), 1–16. 
  7. Lee, D. W., On multiple Appell polynomials, Proc. Amer. Math. Soc. 139, no. 6 (2011), 2133–2141. 
  8. 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. 
  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. 
  10. Luo, Q.-M., Apostol–Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10, no. 4 (2006), 917–925. 
  11. Milne-Thomson, L. M., The Calculus of Finite Differences, Macmillan and Co., Ltd., London, 1951. 
  12. Nørlund, N. E., Differenzenrechnung, Springer-Verlag, Berlin, 1924. 
  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. 
  14. Sandor, J., Crstici, B., Handbook of number theory II, Kluwer Academic Publishers, Dordrecht, 2004. 
  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. 
  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. 
  17. Ward, M., A calculus of sequences, Amer. J. Math. 58 (1936), 255–266. 

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