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Co-recursive associated Jacobi polynomials. (English) Zbl 0828.42013
Starting from the three term recurrence relation for a sequence of orthogonal polynomials $$p_{n+ 2}(x)= (x- \beta_{n+ 1}) p_{n+ 1}(x)- \gamma_{n+ 1} p_n(x)\ (n\ge 0),\ p_0(x)= 1,\ p_1(x)= x- \beta_0$$ with $\beta_n\in {\cal C}$, $\gamma_n\in {\cal C}\backslash \{0\}$, the co-recursive associated polynomials are defined by shifting the index $n$ to $n+ c$ in $\beta_n$, $\gamma_n$ $(n\ge 0)$ and replacing $\beta_0$ by $\beta_0+ \nu$. The author studies the case of the Jacobi polynomials and gives for the co-recursive associated polynomials explicit representations, the orthogonality measure, a fourth order differential equation and he moreover treats 9 limiting cases (including the Laguerre case limit).

MSC:
42C05General theory of orthogonal functions and polynomials
33C05Classical hypergeometric functions, ${}_2F_1$
33C45Orthogonal polynomials and functions of hypergeometric type
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References:
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