By Benz W.

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Rodrigues-type formula. ω(x; a, b, c, d)pn (x; a, b, c, d) = (−1)n n! δ δx n ω(x; a + 12 n, b + 12 n, c + 12 n, d + 12 n) . 10) Generating functions. ∞ 1 F1 a + ix − it a+c 1 F1 d − ix it b+d = 1 F1 a + ix − it a+d 1 F1 c − ix it b+c = (1 − t)1−a−b−c−d 3 F2 1 2 (a pn (x; a, b, c, d) n t . 11) ∞ pn (x; a, b, c, d) n t . 12) + b + c + d − 1), 12 (a + b + c + d), a + ix 4t − a + c, a + d (1 − t)2 ∞ = (a + b + c + d − 1)n p (x; a, b, c, d)tn . n n (a + c) (a + d) i n n n=0 References. [41], [43], [67], [68], [76], [205], [260], [274], [299], [301], [303].

X or equivalently d −x α−1 (α−1) e−x xα L(α) x Ln+1 (x). n (x) = (n + 1)e dx Rodrigues-type formula. e−x xα L(α) n (x) = 1 n! 8) n d dx e−x xn+α . 9) Generating functions. (1 − t)−α−1 exp et 0 F1 (1 − t)−γ 1 F1 − − xt α+1 γ xt α+1 t−1 ∞ xt t−1 n L(α) n (x)t . 10) n=0 ∞ (α) Ln (x) n t . 11) (γ)n n L(α) n (x)t , γ arbitrary. 12) = ∞ = Remarks. 1) of the Laguerre polynomials can also be written as : Ln(α) (x) = 1 n! n k=0 (−n)k (α + k + 1)n−k xk . k! In this way the Laguerre polynomials can be defined for all α.

3 Continuous dual Hahn Definition. Sn (x2 ; a, b, c) = 3 F2 (a + b)n (a + c)n −n, a + ix, a − ix 1 . 1) Orthogonality. If a,b and c are positive except possibly for a pair of complex conjugates with positive real parts, then ∞ 1 2π Γ(a + ix)Γ(b + ix)Γ(c + ix) Γ(2ix) 2 Sm (x2 ; a, b, c)Sn (x2 ; a, b, c)dx 0 = Γ(n + a + b)Γ(n + a + c)Γ(n + b + c)n! δmn . 2) If a < 0 and a + b, a + c are positive or a pair of complex conjugates with positive real parts, then ∞ 1 2π Γ(a + ix)Γ(b + ix)Γ(c + ix) Γ(2ix) 2 Sm (x2 ; a, b, c)Sn (x2 ; a, b, c)dx 0 + Γ(a + b)Γ(a + c)Γ(b − a)Γ(c − a) Γ(−2a) 29 (2a)k (a + 1)k (a + b)k (a + c)k (−1)k (a)k (a − b + 1)k (a − c + 1)k k!