A. G. Shannon

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 9, 2003, Number 4, Pages 73—82

**Download full paper: PDF, 101 Kb**

## Details

### Authors and affiliations

A. G. Shannon

*Warrane College, The University of New South Wales, 1465,
& KvB Institute of Technology, North Sydney, 2060, Australia
*

### Abstract

Generalizations of the polynomials of Bernoulli, Euler and Hermite are defined here in terms of generalized integers called Fermatian integers. These are closely related to the q-series extensively studied by Leonard Carlitz. These various analogues of the classical special functions are inter-related with one another and also to some of the problems posed by Morgan Ward. The works of Henry Gould and Vern Hoggatt are also extensively cited.

### AMS Classification

- 11B65
- 11B68
- 11B39

### References

- Carlitz, An Analogue of the Bernoulli Polynomials,
*Duke Mathematical Journal,***8**(1941): 405-412. - Carlitz, Generalized Bernoulli and Euler Numbers,
*Duke Mathematical Journal,***8**(1941): 585-589. - Carlitz, q-Bernoulli Numbers and Polynomials,
*Duke Mathematical Journal,***15**(1948): 987-1000. - Carlitz, A Note on Hermite Polynomials,
*American Mathematical Monthly,***62**(1955): 646-647. - Carlitz, The Product of Certain Polynomials Analogous to the Hermite Polynomials,
*American Mathematical Monthly,***64**(1957): 723-725. - Carlitz, Expansions of q-Bernoulli Numbers,
*Duke Mathematical Journal,***25**(1958): 355-364. - Carlitz, A Property of the Bernoulli Numbers,
*American Mathematical Monthly,***66**(1959): 714-715. - Carlitz, Some Congruences Involving Binomial Coefficients,
*Elemente der Mathematik,***14**(1959): 11-13. - Carlitz, Congruence Properties of Hermite and Laguerre Polynomials,
*Archiv der Mathematik,***10**(1959): 460-465. - Carlitz, Some Arithmetic Properties of Generalized Bernoulli Numbers,
*Bulletin of the American Mathematical Society,***65**(1959): 68-69. - Carlitz, Arithmetic Properties of Generalized Bernoulli Numbers,
*Journal für die Reine und Angewandte Mathematik,***202**(1959): 174-182. - Carlitz, Some Integral Equations Satisfied by the Complete Elliptic Integrals of the First and Second Kind,
*Bolletino della Unione Matematica Italiana,***(3) 16**(1961): 264-268. - Carlitz, Extended Bernoulli and Eulerian Numbers,
*Duke Mathematical Journal,***31**(1964): 667-690. - Carlitz, Recurrences for the Bernoulli and Euler Numbers,
*Journal für die Reine und Angewandte Mathematik,***214/215**(1964): 184-191. - Carlitz, Bernoulli Numbers,
*The Fibonacci Quarterly,***6**(1968): 71-85. - Carlitz, Generating Functions,
*The Fibonacci Quarterly*,**7**(1969): 359-393. - W. Gould, Stirling Number Representation Problems,
*Proceedings of the American Mathematical Society,***11**(1960): 447-451. - W. Gould, Generating Functions for Powers of Products of Fibonacci Numbers,
*The Fibonacci Quarterly*,**1(2)**(1963): 1-16. - E. Hoggatt Jr, Fibonacci Numbers and Generalized Binomial Coefficients,
*The Fibonacci Quarterly,***5**(1967): 383-400. - F Horadam, Basic Properties of a Certain Generalized Sequence of Numbers,
*The Fibonacci Quarterly***3**(1965): 161-176. - F. Horadam & A. G. Shannon, Ward’s Staudt-Clausen Theorem,
*Mathematica Scandinavica*,**29**(1976): 239-250. - W. Leopoldt, Eine Verallgemeinerung der Bernoullischen Zahlen,
*Abh. Math. Sem. Unive. Hamburg,***22**(1958): 131-140. - E. Nörlund,
*Vorlesungen über Differenzenrechnung.*Berlin: Springer. - Riordan, Abel Identities and Inverse Relations,
*Combinatorial Mathematics and Its Applications.*Chapel Hill: University of North Carolina Press, 1969, pp.71-92. - G. Shannon, Generalized Bernoulli Polynomials and Jackson’s Calculus of Sequences,
*Notes on Number Theory & Discrete Mathematics*,**9**(2003): 1-6. - G. Shannon & A. F. Horadam, Reciprocals of Generalized Fibonacci Numbers,
*The Fibonacci Quarterly*,**9**(1971): 299-306. - N. Vorob’ev,
*Fibonacci Numbers*. Oxford: Pergamon. - Ward, A Calculus of Sequences,
*American Journal of Mathematics*,**58**(1936): 255-266.

## Related papers

## Cite this paper

Shannon, A. G. (2003). Some Fermatian special functions, Notes on Number Theory and Discrete Mathematics, 9(4), 73-82.