Hypergeometric Orthogonal Polynomials

Hypergeometric Orthogonal Polynomials

2010 | R. Koekoek et al.
This chapter focuses on the hypergeometric orthogonal polynomials within the Askey scheme, detailing their most significant properties for each family. These properties include representations as hypergeometric functions, orthogonality relations, three-term recurrence relations, second-order differential or difference equations, forward and backward shift operators, Rodrigues-type formulas, and generating functions. The chapter uses common notations from the literature and highlights connections between different families through limit relations. For the Wilson polynomials, a specific hypergeometric representation is provided: \[ \frac{W_{h}(x^{2} ; a, b, c, d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}} =_{4} F_{3}(\begin{array}{c} -n, n+a+b+c+d-1, a+i x, a-i x \\ a+b, a+c, a+d \end{array} ; 1). \]This chapter focuses on the hypergeometric orthogonal polynomials within the Askey scheme, detailing their most significant properties for each family. These properties include representations as hypergeometric functions, orthogonality relations, three-term recurrence relations, second-order differential or difference equations, forward and backward shift operators, Rodrigues-type formulas, and generating functions. The chapter uses common notations from the literature and highlights connections between different families through limit relations. For the Wilson polynomials, a specific hypergeometric representation is provided: \[ \frac{W_{h}(x^{2} ; a, b, c, d)}{(a+b)_{n}(a+c)_{n}(a+d)_{n}} =_{4} F_{3}(\begin{array}{c} -n, n+a+b+c+d-1, a+i x, a-i x \\ a+b, a+c, a+d \end{array} ; 1). \]
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