### A fast algorithm for the construction of recurrence relations for modified moments

A new approach is presented for constructing recurrence relations for the modified moments of a function with respect to the Gegenbauer polynomials.

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A new approach is presented for constructing recurrence relations for the modified moments of a function with respect to the Gegenbauer polynomials.

Let ${P}_{k}$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients ${a}_{k}$ in $f={\sum}_{k}{a}_{k}{P}_{k}$. A systematic use of the basic properties (including some nonstandard ones) of the polynomials ${P}_{k}$ results in obtaining a low order of the recurrence.

The author reviews results (without proofs) from the theory of minimal projective operators. As he remarks, an excellent introduction to this theory is the survey paper by E. W. Cheney and K. H. Price [Approximation theory (Proc. Sympos., Lancaster, 1969), pp. 261–289, Academic Press, London, 1970; MR0265842]. The author is motivated by a number of papers in this topic published after 1970, bringing essentially new results, e.g., an existence theorem for the minimal operators in the class of all...

Let ${P}_{k}$ be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in ${P}_{i}{P}_{j}={\sum}_{k}c(i,j,k){P}_{k}$, in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by ${P}_{k}$.

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