Titlebook: Hypergeometric Orthogonal Polynomials and Their q-Analogues; Roelof Koekoek,Peter A. Lesky,René F. Swarttouw Book 20101st edition Springer

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Orthogonal Polynomial Solutions in ,, of ,-Difference EquationsIn the case that .=0, .>0 and .≠1 we might replace . by .. in (3.2.1) and then replace . by ... Then we have . In this case the eigenvalue problem reads (cf. (10.1.1)) . for .=0,1,2,…. This can also be written in the symmetric form . for .=0,1,2,…, with . and . The regularity condition (2.3.3) implies that .≠0.

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Orthogonal Polynomial Solutions in , of Complex ,-Difference EquationsIt is also possible to obtain real polynomial solutions of the (complex) .-difference equation (12.2.1) . with argument . where .∈ℝ∖{0} and .,.∈ℂ∖{0}. By using .=.+., .=.+.. with .,.,.,.∈ℝ, we find that the imaginary part of . equals . This is equal to zero for all .∈ℝ and .∈ℝ if

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https://doi.org/10.1007/978-3-642-05014-5Askey scheme; Eigenvalue; Hypergeometric function; basic hypergeometric functions; differential equation

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Orthogonal Polynomial Solutions in ,,+,, of Real ,-Difference Equationsdifferent. This implies by using theorem 3.7 that there exists a sequence of dual polynomials. In this case we have . with .=0 and ..=1=... Furthermore we have by using (11.2.2) . if we choose .=−1 in (11.2.1).

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Definitions and Miscellaneous Formulas,s function .(.) is constant between its (countably many) jump points then we have the situation of positive weights .. on a countable subset . of ℝ. Then the system . is orthogonal on . with respect to these weights as follows:

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