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Abelian groups, module theory, and topology: proceedings in by Dikran Dikranjan, Luigi Salce

By Dikran Dikranjan, Luigi Salce

Encompasses a stimulating choice of papers on abelian teams, commutative and noncommutative jewelry and their modules, and topological teams. Investigates presently well known issues corresponding to Butler teams and nearly thoroughly decomposable teams.

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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!

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