Algebraic Properties of a Family of Generalized Laguerre Polynomials
نویسندگان
چکیده
منابع مشابه
Algebraic Properties of a Family of Generalized Laguerre Polynomials
We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that L n (x) = Pn j=0 `n− j+r n− j ́ x / j! is a Q-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950’s by Grosswald and proven recently by Filaseta and T...
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Three specializations of a set of orthogonal polynomials with “8 different q’s” are given. The polynomials are identified as q-analogues of Laguerre polynomials, and the combinatorial interpretation of the moments give infinitely many new Mahonian statistics on permutations.
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The one-parameter family of polynomials Jn(x, y) = ∑n j=0 ( y+j j ) x is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each n ≥ 6, the polynomial Jn(x, y0) is irreducible over Q for all but finitely many y0 ∈ Q. If n is odd, then with the exception of a finite set of y0, the Galois group of Jn(x, y0) is Sn; if n is even, then the exceptional set is thin.
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belong to one of the three family of orthogonal polynomials, the other two being Jacobi and Legendre. In addition to their important roles in mathematical analysis, these polynomials also feature prominently in algebra and number theory. Schur ([7], [8]) pioneered the study of Galois properties of specializations of these orthogonal polynomials, and Feit [1] used them to solve the inverse Galoi...
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General classes of two variables Appell polynomials are introduced by exploiting properties of an iterated isomorphism, related to the so called Laguerre-type exponentials. Further extensions to the multi-index and multivariable cases are mentioned. 2000 Mathematics Subject Classification. 33C45, 33C99, 30D05, 33B10.
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ژورنال
عنوان ژورنال: Canadian Journal of Mathematics
سال: 2009
ISSN: 0008-414X,1496-4279
DOI: 10.4153/cjm-2009-031-6