Polynomial Expansions of Analytic Functions
Reihe: Moderne Funktionentheorie
Published on 1. January 1964
Book
Hardback
978-3-540-03123-9 (ISBN)
Description
This monograph deals with the expansion properties, in the complex domain, of sets of polynomials which are defined by generating relations. It thus represents a synthesis of two branches of analysis which have been developing almost independently. On the one hand there has grown up a body of results dealing with the more or less formal prop erties of sets of polynomials which possess simple generating relations. Much of this material is summarized in the Bateman compendia (ERDELYI [1J, vol. III, chap. 19) and in TRUESDELL [1J. On the other hand, a problem of fundamental interest in classical analysis is to study the representability of an analytic function j(z) as a series 2::CnPn(z), where {Pn} is a prescribed sequence of functions, and the connections between the function j and the coefficients en. BIEBERBACH'S mono graph Analytisehe Fortsetzung (Ergebnisse der Mathematik, new series, no. 3) can be regarded as a study of this problem for the special choice Pn (z) = zn, and illustrates the depth and detail which such a specializa tion allows. However, the wealth of available information about other sets of polynomials has seldom been put to work in this connection (the application of generating relations to expansion of functions is not even mentioned in the Bateman compendia). At the other extreme, J. M.
More details
Series
Edition
1st ed. 1958. Corr. 2nd printing
Language
English
Place of publication
Heidelberg
Germany
Publishing group
Springer Berlin
Target group
Research
Product notice
sewn/stitched
Cloth over boards
Weight
245 gr
ISBN-13
978-3-540-03123-9 (9783540031239)
DOI
10.1007/978-3-642-87887-9
Schweitzer Classification
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Ralph P. Boas | Robert Creighton Buck
Polynomial expansions of analytic functions
Book
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Content
I. Introduction.- § 1. Generalities.- § 2. Representation formulas with a kernel.- § 3. The method of kernel expansion.- § 4. Lidstone series.- § 5. A set of Laguerre polynomials.- § 6. Generalized Appell polynomials.- II. Representation of entire functions.- § 7. General theory.- § 8. Multiple expansions.- § 9. Appell polynomials.- (i) Bernoulli polynomials and generalizations.- (ii) A set of Laguerre polynomials.- (iii) Hermite polynomials.- (iv) Reversed Laguerre polynomials.- (v) Reversed Rainville polynomials.- § 10. Sheffer polynomials.- (vi) General difference polynomials.- (vii) Poisson-Charlier, Narumi and Boole polynomials.- (viii) Mittag-Leffier polynomials.- (ix) Abel interpolation series.- (x) Laguerre polynomials.- (xi) Angelescu polynomials.- (xii) Denisyuk polynomials.- (xiii) Squared Hermite polynomials.- (xiv) Adhoc polynomials.- (xv) Actuarial polynomials.- § 11. More general polynomials.- (xvi) Special hypergeometric polynomials.- (xvii) Reversed Bessel polynomials.- (xviii) q-difference polynomials.- (xix) Reversed Hermite polynomials.- (xx) Rainville polynomials.- § 12. Polynomials not in generalized Appell form.- III. Representation of functions that are regular at the origin.- § 13. Integral representations.- § 14. Brenke polynomials.- (i) Polynomials generated by A(?) (1 - z?)-?.- (ii) q-difference polynomials.- § 15. More general polynomials.- § 16. Polynomials generated by A(?) (1 - zg(?))-?.- (iii) Taylor series.- (iv) Lerch polynomials.- (v) Gegenbauer polynomials.- (vi) Chebyshev polynomials.- (vii) Humbert polynomials.- (viii) Faber polynomials.- § 17. Special hypergeometric polynomials.- (ix) Jacobi polynomials.- § 18. Polynomials not in generalized Appell form.- IV. Applications.- § 19. Uniqueness theorems.- § 20. Functional equations.