By I. Dolgachev

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GEOMETRY. --- DEFINITIONS. 1. If a block of wooden or stone be minimize within the sbape represented iu Fjg. 1, it is going to have six fiat faces. every one face of the block is named a floor; and if those faces are made gentle by way of sharpening, in order that, whilst a straight-edge is utilized to anybody of them, the instantly facet in everything will contact the skin, the faces are referred to as aircraft surfaces, or planes.

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N }. 1) in the following way : p Kn (x; p, N ) = Mn x; −N, . p−1 References. [13], [31], [32], [39], [43], [50], [64], [67], [104], [119], [123], [136], [142], [145], [146], [154], [159], [181], [183], [212], [250], [272], [274], [286], [287], [294], [296], [298], [301], [307], [323], [338], [340], [385], [386], [388], [407], [409]. 11 Laguerre Definition. L(α) n (x) = (α + 1)n 1 F1 n! 47 −n x . 1) Orthogonality. ∞ (α) e−x xα L(α) m (x)Ln (x)dx = Γ(n + α + 1) δmn , α > −1. n! 2) 0 Recurrence relation.

Remarks. The Hermite polynomials can also be written as : Hn (x) = n! [n/2] k=0 (−1)k (2x)n−2k , k! (n − 2k)! where [α] denotes the largest integer smaller than or equal to α. 1) are connected by the following quadratic transformations : (− 12 ) H2n (x) = (−1)n n! 22n Ln and (x2 ) (1) H2n+1 (x) = (−1)n n! 22n+1 xLn2 (x2 ). References. [2], [10], [13], [18], [19], [31], [34], [39], [43], [49], [64], [74], [82], [87], [89], [91], [92], [112], [123], [128], [131], [137], [138], [154], [158], [195], [196], [200], [201], [202], [214], [239], [274], [285], [288], [301], [302], [306], [314], [316], [323], [329], [332], [360], [367], [376], [381], [388], [390], [394], [397], [406].

N=0 if β + δ + 1 = −N or γ + 1 = −N. 11) −x, −x + β − γ t β+δ+1 2 F1 x + α + 1, x + γ + 1 t α−δ+1 2 F1 N = (α + 1)n (γ + 1)n Rn (λ(x); α, β, γ, δ)tn , (α − δ + 1) n! n n=0 if α + 1 = −N or γ + 1 = −N. 12) x + α + 1, x + β + δ + 1 t α+β−γ+1 N = (α + 1)n (β + δ + 1)n Rn (λ(x); α, β, γ, δ)tn , (α + β − γ + 1) n! n n=0 if α + 1 = −N or β + δ + 1 = −N. 13) N N (α + β + 1)n Rn (λ(x); α, β, γ, δ)tn . = n! 14) Remark. 1) : Rn (λ(−a + ix); a + b − 1, c + d − 1, a + d − 1, a − d) Wn (x2 ; a, b, c, d) ˜ n (x2 ; a, b, c, d) = .