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Ralph P.Jr. Boas, R.C. Buck, "Polynomial Expansions of Analytic Functions"

Contents

Chapter I. Introduction 1

§ 1. Generalities 1

§ 2. Representation formulas with a kernel 4

§ 3. The method of kernel expansion 10

§ 4. Lidstone series 13

§ 5. A set of Laguerre polynomials 16

§ 6. Generalized Appell polynomials 17

Chapter II. Representation of entire functions 21

§ 7. General theory 21

§ 8. Multiple expansions 24

§ 9. Appell polynomials 28

(i) Bernoulli polynomials and generalizations 29

(ii) A set of Laguerre polynomials 31

(iii) Hermite polynomials 31

(iv) Reversed Laguerre polynomials 32

(v) Reversed Rainville polynomials 32

§ 10. Sheffer polynomials 33

(vi) General difference polynomials 34

(vii) Poisson-Charlier, Narumi and Boole polynomials 37

(viii) Mittag-Leffler polynomials 38

(ix) Abel interpolation series 38

(x) Laguerre polynomials 40

(xi) Angelescu polynomials 41

(xii) Denisyuk polynomials 41

(xiii) Squared Hermite polynomials 41

(xiv) Adhoc polynomials 41

(xv) Actuarial polynomials 42

§ 11. More general polynomials 42

(xvi) Special hypergeometric polynomials 43

(xvii) Reversed Bessel polynomials 43

(xviii) q-difference polynomials 44

(xix) Reversed Hermite polynomials 45

(xx) Rainville polynomials 46

§ 12. Polynomials not in generalized Appell form 46

Chapter III. Representation of functions that are regular at the origin 47

§ 13. Integral representations 47

§ 14. Brenke polynomials 51

(i) Polynomials generated by etc. 52

(ii)q-difference polynomials 54

§ 15. More general polynomials 55

§ 16. Polynomials generated by etc. 57

(iii) Taylor series 57

(iv) Lerch polynomials 57

(v) Gegenbauer polynomials 58

(vi) Chebyshev polynomials 58

(vii) Humbert polynomials 58

(viii) Faber polynomials 59

§ 17. Special hypergeometric polynomials 60

(ix) Jacobi polynomials 60

§ 18. Polynomials not in generalized Appell form 61

ChapterIV. Applications 65

§ 19. Uniqueness theorems 65

§ 20. Functional equations 67

Bibliography 71

Index 75

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