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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).

42C05General theory of orthogonal functions and polynomials
33C05Classical hypergeometric functions, ${}_2F_1$
33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
[1] Bailey, W. N.: Generalized hypergeometric series. (1972) · Zbl 0011.02303
[2] Chihara, T. S.: On co-recursive orthogonal polynomials. Proc. amer. Math. soc. 8, 899-905 (1957) · Zbl 0080.27305
[3] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.: 2nd russian ed. Higher transcendental functions. Higher transcendental functions (1953)
[4] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G.: 2nd russian ed. Higher transcendental functions. Higher transcendental functions (1953)
[5] Hahn, W.: Über orthogonalpolynome mit drei parametern. 5 (1940 1941) · Zbl 66.0314.02
[6] Ismail, M. E. H.; Letessier, J.; Valent, G.; Wimp, J.: Two families of associated Wilson polynomials. Canad. J. Math. 42, 659-695 (1990) · Zbl 0712.33005
[7] Ismail, M. E. H.; Masson, D. R.: Two families of orthogonal polynomials related to Jacobi polynomials. Rocky mountain J. Math. 21, 359-375 (1991) · Zbl 0744.33004
[8] Letessier, J.: On co-recursive associated Laguerre polynomials. J. comput. Appl. math. 49, 127-136 (1993) · Zbl 0792.33005
[9] Luke, Y. L.: 2nd russian ed. The special functions and their approximations. The special functions and their approximations (1969) · Zbl 0193.01701
[10] Orr, W. Mcf.: On the product $Jm(x)Jn(x)$. Proc. Cambridge philos. Soc. 10, 93-100 (1990) · Zbl 30.0417.01
[11] Ronveaux, A.; Marcellán, F.: Co-recursive orthogonal polynomials and fourth-order differential equation. J. comput. Appl. math. 25, No. 1, 105-109 (1989) · Zbl 0662.33005
[12] Shohat, J. A.; Tamarkin, J. D.: The problem of moments. Math. surveys 1 (1950) · Zbl 0041.43302
[13] Wimp, J.: Explicit formulas for the associated Jacobi polynomials and some applications. Canad. J. Math. 39, 983-1000 (1987) · Zbl 0643.33009