Softcover ISBN:  9780821850831 
Product Code:  CONM/75 
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Electronic ISBN:  9780821876640 
Product Code:  CONM/75.E 
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Book DetailsContemporary MathematicsVolume: 75; 1988; 122 ppMSC: Primary 05; Secondary 33;
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 coworkers 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

Chapters

1. Introduction

2. Definitions and Preliminary Results

3. The vUmbral Algebra

4. The vUmbral Field

5. The Group of vDelta Functionals Under Composition

6. Generalized Binomial Polynomial Sequences

7. The Composition of Polynomial Sequences

8. Compositions of Moebius Delta Functionals

9. Generalized Shift Invariant Operators

10. The Generalized Derivative of vShift Invariant Operators

11. Generalized Sheffer Polynomials

12. Cross Sets of Polynomials

13. A Class of Laguerre Type Polynomials

14. The Generalized Heat Polynomials

15. A Primitive Integral for the Euler Operator

16. Bernoulli Type Polynomials and Numbers

17. Generalized Euler Polynomials and Numbers

18. Generalized Stirling Numbers and Factor Polynomials

Bibliography


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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 coworkers 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.

Chapters

1. Introduction

2. Definitions and Preliminary Results

3. The vUmbral Algebra

4. The vUmbral Field

5. The Group of vDelta Functionals Under Composition

6. Generalized Binomial Polynomial Sequences

7. The Composition of Polynomial Sequences

8. Compositions of Moebius Delta Functionals

9. Generalized Shift Invariant Operators

10. The Generalized Derivative of vShift Invariant Operators

11. Generalized Sheffer Polynomials

12. Cross Sets of Polynomials

13. A Class of Laguerre Type Polynomials

14. The Generalized Heat Polynomials

15. A Primitive Integral for the Euler Operator

16. Bernoulli Type Polynomials and Numbers

17. Generalized Euler Polynomials and Numbers

18. Generalized Stirling Numbers and Factor Polynomials

Bibliography