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- W1972188214 abstract "Let ( R [x], 〈c〉) be the algebra of all polynomials in the indeterminate x over the field of real numbers R , where 〈 c 〉 R [ x ] × R → R [ x ] → [ x ] is the convolution product defined by x n 〈 c 〉 x m = n! m! x n + m (n + m)! . We call a polynomial sequence { q n ( x )} n = 0 ∞ a c -orthogonal polynomial sequence if there exists a linear functional L : R 〈 x 〉 → R , such that for all non-negative integers n and m , L ( q n ( x ) 〈 c 〉 q m ( x )) = k n δ n , m where k n ≠ 0, for n = 0, 1, 2,…. The main result of this paper is to show that ψ maps every c -orthogonal polynomial sequence to a c -orthogonal polynomial sequence if and only if ψ is defined by ψ( x n n! ) = sb n L n ( − ax b ), n = 0, 1,2,…, where s , a , and b belong to R , s ≠ 0, a ≠ 0, and { L n ( x )} n = 0 ∞ is the Laguerre polynomial sequence of order zero. Thus, we call the linear operator γ a , b : R [ x ] → R [ x ] defined by γ a, b ( x n n! ) = b n L n ( − ax b ) the convolution shift and obtain some results involving the Laguerre polynomial sequence that are suggested by the analogous results for the ordinary shift operator τ a , b : R [ x ] → R [ x ], defined by τ a , b ( x n ) = ( ax + b ) n ." @default.
- W1972188214 created "2016-06-24" @default.
- W1972188214 creator A5062070377 @default.
- W1972188214 date "1991-05-01" @default.
- W1972188214 modified "2023-10-14" @default.
- W1972188214 title "Convolution shift, c-orthogonality preserving maps, and the Laguerre polynomials" @default.
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- W1972188214 doi "https://doi.org/10.1016/0022-247x(91)90150-x" @default.
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