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1) are related in the following way : (β)n Mn (x; β, c) = Pn(β−1,−n−β−x) n! 2−c c . 1) in the following way : p Kn (x; p, N ) = Mn x; −N, . p−1 References. [6], [10], [13], [19], [21], [31], [32], [39], [43], [50], [52], [64], [67], [69], [80], [104], [123], [130], [154], [170], [172], [173], [181], [183], [212], [222], [227], [233], [239], [247], [250], [274], [286], [287], [296], [298], [301], [307], [316], [323], [338], [391], [394], [407], [409]. 10 Krawtchouk Definition. Kn (x; p, N ) = 2 F1 −n, −x 1 −N p , n = 0, 1, 2, .

N n! References. [6], [10], [13], [19], [21], [31], [32], [39], [50], [64], [67], [81], [123], [124], [142], [154], [181], [183], [212], [222], [274], [286], [287], [288], [294], [296], [298], [301], [307], [316], [323], [388], [394], [407], [409]. 13 Hermite Definition. Hn (x) = (2x)n 2 F0 Orthogonality. −n/2, −(n − 1)/2 1 − 2 − x . 1) ∞ 1 √ π 2 e−x Hm (x)Hn (x)dx = 2n n! δmn . 2) −∞ Recurrence relation. Hn+1 (x) − 2xHn (x) + 2nHn−1 (x) = 0. 3) Normalized recurrence relation. 4) where Hn (x) = 2n pn (x).

41) or equivalently d 1 − x2 dx Rodrigues-type formulas. 1 2 Un (x) = −(n + 1) 1 − x2 1 (1 − x2 )− 2 Tn (x) = 1 (1 − x2 ) 2 Un (x) = (−1)n ( 12 )n 2n d dx (n + 1)(−1)n ( 32 )n 2n Generating functions. − 12 Tn+1 (x). 42) n d dx 1 (1 − x2 )n− 2 . 43) n 1 (1 − x2 )n+ 2 . 44) ∞ 1 − xt = Tn (x)tn . 45) ∞ R − (x − 1)t 1 2 2 0 F1 γ, −γ 1 − R − t 1 2 2 ∞ − (x + 1)t 1 2 2 0 F1 2 F1 = n=0 1 − 2xt + t2 . 46) Tn (x) n t . 1 2 n n! 47) ∞ − (x2 − 1)t2 1 4 2 ext 0 F1 2 F1 1 1 2 n (1 + R − xt) = Tn (x)tn , R = 2 n!

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