**Contemporary Mathematics**

Volume: 75;
1988;
122 pp;
Softcover

MSC: Primary 05;
Secondary 33

Print ISBN: 978-0-8218-5083-1

Product Code: CONM/75

List Price: $31.00

AMS Member Price: $24.80

MAA Member Price: $27.90

**Electronic ISBN: 978-0-8218-7664-0
Product Code: CONM/75.E**

List Price: $31.00

AMS Member Price: $24.80

MAA Member Price: $27.90

# The Finite Calculus Associated with Bessel Functions

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*Frank M. Cholewinski*

Although Bessel functions are among the most widely used functions in applied mathematics, this book is essentially the first to present a calculus associated with this class of functions. The author obtains a generalized umbral calculus associated with the Euler operator and its associated Bessel eigenfunctions for each positive value of an index parameter. For one particular value of this parameter, the functions and operators can be associated with the radial parts of \(n\)-dimensional Euclidean space objects. Some of the results of this book are in part extensions of the work of Rota and his co-workers on the ordinary umbral calculus and binomial enumeration. The author also introduces a wide variety of new polynomial sequences together with their groups and semigroup compositional properties. Generalized Bernoulli, Euler, and Stirling numbers associated with Bessel functions and the corresponding classes of polynomials are also studied. The book is intended for mathematicians and physicists at the research level in special function theory.

# Table of Contents

## The Finite Calculus Associated with Bessel Functions

- Contents xi12 free
- 1. Introduction 114 free
- 2. Definitions and Preliminary Results 215
- 3. The v-Umbral Algebra 1225
- 4. The v-Umbral Field 1831
- 5. The Group of v-Delta Functionals Under Composition 2134
- 6. Generalized Binomial Polynomial Sequences 2639
- 7. The Composition of Polynomial Sequences 3548
- 8. Compositions of Molebius Delta Functionals 4154
- 9. Generalized Shift Invariant Operators 4457
- 10. The Generalized Derivative of v-Shift Invariant Operators 5063
- 11. Generalized Sheffer Polynomials 5972
- 12. Cross Sets of Polynomials 7386
- 13. A Class of Laguerre Type Polynomials 8093
- 14. The Generalized Heat Polynomials 8699
- 15. A Primitive Integral for the Euler Operator 88101
- 16. Bernoulli Type Polynomials and Numbers 91104
- 17. Generalized Euler Polynomials and Numbers 102115
- 18. Generalized Stirling Numbers and Factor Polynomials 109122
- Bibliography 120133