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Handbook of mathematics for engineers and scienteists part 146. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 18.17. Orthogonal Polynomials 983 18.17.1-2. Generalized Laguerre polynomials. The generalized Laguerre polynomials L La x a -1 satisfy the equation xyXX a 1 - x y x ny 0 and are defined by the formulas 7- . . 1 -a x dn n a -x r n-m x m T n a 1 -x m La x x ae xn ae x C2 7 77. n dxn 7 m T m a 1 m n-m m 0 m 0 Notation L x Ln x . Special cases Lo x 1 L x a 1 -x L-nn x - 1 n f. To calculate L x for n 2 one can use the recurrence formulas L 1 æ y 2n a 1 - x L x - n a L0_ 1 x . Other recurrence formulas a a a 1 aa 1 at t a a Ln x Ln-1 x Ln x dxL n X Ln-1 x X dxL n X Ln x n a Ln- 1 x . The functions La x form an orthogonal system on the interval 0 x œ with weight x a e x xae xl.fixdf x dx 0 t JO Ln The generating function is n m n m. if if œ 1 -s a-1 exp - La x sn n 0 s 1. 18.17.2. Chebyshev Polynomials and Functions 18.17.2-1. Chebyshev polynomials of the first kind. The Chebyshev polynomials of the first kind Tn Tn x satisfy the second-order linear ordinary differential equation 1 -x2 yXX -xyX n2y 0 18.17.2.1 and are defined by the formulas -2 nn ------r d 2 n_i Tn x cos n arccos x 2 v1 -x2 -dp 1 -x2 2 n 2 ne -Dm n tm. 1 v 2x n-2m n 0 1 2 . 2 m n- 2m m 0 where A stands for the integer part of a number A. 984 Special Functions and Their Properties An alternative representation of the Chebyshev polynomials T x 1 ra i - f 2 1 - x2 n-1 2 C - 1 1 X dxn 1 X The first five Chebyshev polynomials of the first kind are To x 1 T1 x x T2 x 2x2 - 1 T3 x 4x3 - 3x T4 x 8x4 - 8x2 1. The recurrence formulas Tn 1 x 2xTn x - Tn-1 x n 2. The functions Tn x form an orthogonal system on the interval -1 x 1 with f1 Tn x Tm x J 0 if n m ------- dx 2n if n m 0 - -1 v1 -x In if n m 0. The generating function is tt T- g Tn x Sn 1 - The functions Tn x have only real simple zeros all lying on the interval -1 x 1. The normalized Chebyshev polynomials of the first kind 21-nTn x deviate from zero least of all. This means that among all polynomials of degree n with the leading coefficient 1 it is the .
