Rising factorial Bernoulli polynomials

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 14, 2008, Number 1, Pages 1–5
Full paper (PDF, 38 Kb)

Details

Authors and affiliations

A. G. Shannon
Warrane College, The University of New South Wales, Kensington 1465, &
Raffles College, 99 Mount Street, North Sydney, NSW 2065, Australia

Absrtacts

This paper considers some properties of rising binomial coefficients and two analogs of the Bernoulli polynomials which can be developed from them.

Keywords

• q-series
• Binomial coefficients
• Rising factorials
• Generalized Bernoulli polynomials
• Gauss–Cayley generalizations
• Fermatians

AMS Classification

• 11B65
• 11B39
• 05A30

References

1. Carlitz, L. q-Bernoulli Numbers and Polynomials. Duke Mathematical Journal. 16 (1949): 987-1000.
2. Carlitz, L. Generating Functions. The Fibonacci Quarterly. Vol.7 (1969): 359-393.
3. Dyson, F. A Walk through Ramanujan’s Garden. In George E Andrews et al (eds). Ramanujan Revisited. San Diego, CA: Academic Press, pp.7-28.
4. Ismail, Mourad. Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications, 98. Cambridge: Cambridge University Press.
5. Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, 1958, p.3.
6. Riordan, J. Abel Identities and Inverse Relations. In Combinatorial Mathematics and Its Applications. Chapel Hill, NC: University of North Carolina Press, pp.71-92.
7. Shannon, A.G. Some q-Binomial Coefficients Formed from Rising Factorials. Notes on Number Theory and Discrete Mathematics. 12 (2006): 13-20.
8. Sylvester, J.J. The Collected Mathematical Papers of James Joseph Sylvester. Volume IV (1882-1897). New York: Chelsea, 1973, pp.91,93-94.