Generalized differential operators

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 3, Pages 38–44
Full paper (PDF, 44 Kb)

Details

Authors and affiliations

A. G. Shannon

Faculty of Engineering & IT, University of Technology
Sydney, NSW 2007, Australia

Abstract

This paper considers some properties of generalized differential operators by extending Chak and Schur derivatives as previously investigated by Leonard Carlitz. They are applied in the context of extended Laguerre polynomials.

Keywords

• Rising and falling factorials
• Binomial coefficients
• Chak derivatives
• Schur derivatives
• Laguerre polynomials
• q-series

AMS Classification

• 11B65
• 11B39
• 05A30

References

1. Carlitz, L. q-Bernoulli Numbers and Polynomials. Duke Mathematical Journal. 15, 1948, 987-1000.
2. Carlitz, L. The Schur Derivative of a Polynomial. Proceedings of the Glasgow Mathematical Association. Vol. 1, 1953, 59–163.
3. Carlitz, L. Expansions of q-Bernoulli Numbers. Duke Mathematical Journal. 25, 1958, 355-364.
4. Carlitz, L. Sums of products of Multinomial Coefficients. Elemente der Mathematik. Vol. 18, 1963, 37–39.
5. Carlitz, L. Multiple Sums and Generating Functions. Collectanea Mathematica. Vol. 17, 1965, 281–296.
6. Carlitz, L. Some Generating Functions of Laguerre Polynomials. Duke Mathematical Journal. Vol. 35, 1968, 825–828.
7. Carlitz, L. Generating Functions. The Fibonacci Quarterly. Vol. 7, 1969, 359–393.
8. Carlitz, L., J. Riordan. Two Element Lattice Permutation Numbers and Their q-generalization. Duke Mathematical Journal. Vol. 31, 1964, 371–388.
9. Riordan, J. A Note on a q-extension of Ballot Numbers. Journal of Combinatorial Theory. Vol. 4, 1968, 191–193.
10. Shannon, A. G. Some Properties of Modified Lah Numbers. Notes on Number Theory and Discrete Mathematics. Vol. 7, 2001, No. 4, 125–131.
11. Shannon, A. G. Some q-Binomial Coefficients Formed from Rising Factorials. Notes on Number Theory and Discrete Mathematics. Vol. 12, 2006, No. 1, 13–20.
12. Shannon, A. G. Some Generalized Rising Binomial Coefficients. Notes on Number Theory and Discrete Mathematics. Vol. 13, 2007, No. 1, 25–30.