# Boundary Value Problems for Operator Differential Equations by Myroslav L. Gorbachuk

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Example text

X) is a polynomial of degree n of the form H. (x) = 2nxn +. The polynomials H. (x) satisfy the following relationship: for m#n 0 J00 e-x2Hn (x) Hm (x) dx = 2n\/-,, n! for m = n. ) form an orthogonal system on e_ , 2. 1) by Hn (x) = 2 2 H,, l `1. = 00 0Hm(n)(x)dx. f-W -oo Form < n, Hr(n) (x) = 0 and thus I = 0. If m = n then 00 I= 2nn! 0o 95 dx = 21-n! oo An m e-" dx = 24n1 c. system is formed by the polynomials H. 4) 2or 4. 5. CHEBYSHEV-LAGUERRE POLYNOMIALS The Chebyshev-Laguerre polynomials are orthogonal on the half-line 0 S x < o with respect to the weight function p (x) = xae-".

X) its expression in terms of 40n: (-1)n J 00 0-(a)Lm(a)(x)dx 0 oo (-1)n0n(n-1)Lm(a)(x) + I (-1)n-1 = (-1)n-1 J 0n(n-1)[Lm(a)(x)]' dx = 0 0 oM S6n(n-1)[Lm(a)(x)], dx. 0 The term which does not involve the integral vanishes because a> -1. Carrying out the integration by parts n times we obtain 1= J0 0. [Lm(a)(x)](n)dx. Form < n, we have [L m(a) (x)] (n) = 0 and therefore 1= 0. When m = n, 1 = a! f cdx = n! f xa+nedx = n! I' (a + n + 1). 0 The orthonormal Chebyshev-Laguerre polynomials are Ln(a)(x) ln(a)(x)= 1.

0 The orthonormal Chebyshev-Laguerre polynomials are Ln(a)(x) ln(a)(x)= 1. ]P (a + n + 1)] 2. 5) REFERENCES V. L. Goncharov, Theory of Interpolation and Approximation of Functions, Gostekhizdat, Moscow, 1954, Chap. 3, 4 (Russian). D. Jackson, Fourier Series and Orthogonal Polynomials, Cams Monograph No. 6, Math. Assoc. , 1941, Chap. 2, 7-10. 36 Preliminary Information A. N. Korkin and E. I. " Collected works of E. I. Zolotarev, Vol. 1, pp. 138-153. I. P. Natanson, Constructive Theory of Functions, Gostekhizdat, Moscow, 1949 (Russian). 