Some Fermatian special functions

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 9, 2003, Number 4, Pages 73–82
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

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