Special Functions"Based upon the lectures on special functions which ... (the author has) been giving at the University of Michigan since 1946.". |
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Page 300
... Euler polynomials , and the numbers ( 2 ) E1 = 2 ′′ E , ( 3 ) are called Euler numbers . The polynomials E , ( x ) / n ! are of Sheffer A - type zero . ( 3 ) ( 4 ) ( 5 ) ( 6 ) It is not difficult to obtain such results as En ( x + 1 ) + ...
... Euler polynomials , and the numbers ( 2 ) E1 = 2 ′′ E , ( 3 ) are called Euler numbers . The polynomials E , ( x ) / n ! are of Sheffer A - type zero . ( 3 ) ( 4 ) ( 5 ) ( 6 ) It is not difficult to obtain such results as En ( x + 1 ) + ...
Page 360
... Euler , Leonhard , 1 , 8-9 , 11-12 , 15 , 26-27 , 31 , 47 , 60 , 300 Euler constant , 8-9 Euler integral , Gamma function , 15 , 17 Euler numbers , 300 Euler polynomials , 300 Euler product for Gamma function , 15 , 17 Euler summation ...
... Euler , Leonhard , 1 , 8-9 , 11-12 , 15 , 26-27 , 31 , 47 , 60 , 300 Euler constant , 8-9 Euler integral , Gamma function , 15 , 17 Euler numbers , 300 Euler polynomials , 300 Euler product for Gamma function , 15 , 17 Euler summation ...
Page 361
... Euler integral , 9 , 15–18 Euler product , 11 evaluation of infinite products , 13-15 generalization of factorial , 12 incomplete , 127 limit form , 11-12 multiplication theorem , 26 product form , 9 , 11 relation to Beta function , 18 ...
... Euler integral , 9 , 15–18 Euler product , 11 evaluation of infinite products , 13-15 generalization of factorial , 12 incomplete , 127 limit form , 11-12 multiplication theorem , 26 product form , 9 , 11 relation to Beta function , 18 ...
Contents
INFINITE PRODUCTS 1 Introduction 2 Definition of an infinite product | 1 |
A necessary condition for convergence 4 The associated series of logarithms | 2 |
Absolute convergence Page 1 1 2 | 3 |
Copyright | |
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a₁ absolutely convergent analytic b₁ b₂ basic table Bateman's Bessel functions Bessel polynomials Brafman c)nn Celine's polynomials Chapter coefficients conclude constant contiguous function relations defined derive differential equation differential recurrence relation elliptic function exists exp(2xt F₁ F₁(a factor finite follows formula Gegenbauer polynomials Hence Hermite polynomials hypergeometric function independent infinite product integral Jacobi polynomials Laguerre polynomials left member Legendre polynomials Lemma Math negative integer nomials non-negative integer notation obtain on(x Pn(x poles poly polynomial sets preceding section Proof properties pure recurrence relation Rainville Re(a Re(b replace right member Rodrigues formula satisfied set of polynomials Sheffer A-type zero Show simple set Sister Celine's sn(u solution summation Theorem Theorem 48 theta functions W₁ write yields ακ Σ Σ ΣΣ