By Benjamin Peirce

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**Extra resources for An elementary Treatise on Plane and Solid Geometry**

**Example text**

X! 2) Recurrence relation. (c − 1)xMn (x; β, c) = c(n + β)Mn+1 (x; β, c) − [n + (n + β)c] Mn (x; β, c) + nMn−1 (x; β, c). 3) Normalized recurrence relation. 4) n pn (x). Difference equation. n(c − 1)y(x) = c(x + β)y(x + 1) − [x + (x + β)c] y(x) + xy(x − 1), y(x) = Mn (x; β, c). 5) Forward shift operator. 6) or equivalently ∆Mn (x; β, c) = n β c−1 c Mn−1 (x; β + 1, c). 7) Backward shift operator. 8) or equivalently ∇ (β)x cx (β − 1)x cx Mn (x; β, c) = Mn+1 (x; β − 1, c). x! x! 9) Rodrigues-type formula.

For x = 0, 1, 2, . . , N we have 1 F1 −x −t α+1 1 F1 x−N t β+1 N = (−N )n Qn (x; α, β, N )tn . (β + 1) n! 11) x − N, x + α + 1 t − N = (−N )n (α + 1)n Qn (x; α, β, N )tn . n! 12) N N = (α + β + 1)n Qn (x; α, β, N )tn . n! 13) Remark. 1). 2) of the dual Hahn polynomials : N (2n + α + β + 1)(α + 1)n (−N )n N ! Qn (x; α, β, N )Qn (y; α, β, N ) (−1)n (n + α + β + 1)N +1 (β + 1)n n! n=0 = α+x x δxy , x, y ∈ {0, 1, 2, . . , N }. β+N −x N −x References. [13], [31], [32], [39], [43], [50], [64], [67], [69], [123], [127], [130], [136], [142], [143], [181], [183], [212], [215], [251], [271], [274], [286], [287], [290], [294], [295], [296], [298], [301], [307], [323], [336], [338], [339], [344], [366], [385], [386], [399], [402], [407].

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. 1) Orthogonality. N x=0 N x (−1)n n!