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The Finite Calculus Associated with Bessel Functions
Frank M. Cholewinski
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Contemporary Mathematics
1988; 122 pp; softcover
Volume: 75
ISBN-10: 0-8218-5083-0
ISBN-13: 978-0-8218-5083-1
List Price: US$29
Member Price: US$23.20
Order Code: CONM/75
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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 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 \(\nu\)-umbral algebra
  • The \(\nu\)-umbral field
  • The group of \(\nu\)-delta functionals under composition
  • Generalized binomial polynomial sequences
  • The composition of polynomial sequences
  • Compositions of Moebius delta functionals
  • Generalized shift invariant operators
  • The generalized derivative of \(\nu\)-shift invariant operators
  • Generalized Sheffer polynomials
  • Cross sets of polynomials
  • A class of Laguerre type polynomials
  • The generalized heat polynomials
  • A primitive integral for the Euler operator
  • Bernoulli type polynomials and numbers
  • Generalized Euler polynomials and numbers
  • Generalized Stirling numbers and factor polynomials
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