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If we take the difference of these two equations we get ∞ 0 = 2(m − n) −∞ 2 e−x Rn (x)Rm (x)dx. It follows that since m = n ∞ −∞ 2 e−x Rn (x)Rm (x)dx = 0. 2) This is the orthogonality relation satisfied by the Hermite polynomials. Therefore if we multiply Rn (x) by a constant so that the coefficient of xn is 2n , we will obtain the Hermite polynomial Hn (x). Since the series consists of only even powers of x or only odd powers, the coefficient kn = 0. 4 Derivative Property The p th derivative of Hn (x) is a polynomial of order n − p.
Paris, 200 (1935), 2052-2053. Hochstrasser Urs W, Orthogonal Polynomials, Chapter 22, Handbook of Mathematical Functions, Eds. Abramowitz M and Stegun I A, Dover, 1970. Koornwinder T H, Wong R, Koekoek R and Swarttouw R F, Chapter 18, NIST Handbook of Mathematical Functions, Eds. Olver W J, Lozier D W, Boisvert R F and Clark C W, NIST and Cambridge University Press, 2009. Natanson I, Konstruktive Functionentheorie, Akademie-Verlag, Berlin 1955. Shohat J, Theorie Generale des Polynomes Orthogonaux de Tchebyshef, Memoires des Sciences Mathematiques, 66 (1934) 1-68.
X)k Γ(α + k + 1)(n − k)! k! n−k−m i=0 = i! (n − i − 1)! (n − m − i)! (i + k)! (n − i − k − 1)! (m − 1)! n! Γ(α + m + k + 1) (n − m − i − k)! i! k! (k + 1)(n − m − k) (n − k − 1)! 1+ +... (m − 1)! n! Γ(α + m + k + 1)(n − m − k)! (n − k − 1) page 46 August 21, 2015 8:44 ws-book9x6 World Scientific Book - 9in x 6in 9700-main 47 Associated Laguerre Polynomials The sum inside the curly brackets is the hypergeometric polynomial 2 F1 (k + 1, m + k − n; k + 1 − n) which is most easily evaluated using Vandermonde’s theorem (−n)n−m−k 2 F1 (k + 1, m + k − n; k + 1 − n; 1) = k + 1 − n)n−m−k = (k + 1 + m)n−m−k n!