contiguous relations (Heine)
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1: 15.5 Derivatives and Contiguous Functions
§15.5 Derivatives and Contiguous Functions
… ►§15.5(ii) Contiguous Functions
►The six functions , , are said to be contiguous to . … ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …Further contiguous relations include: …2: 16.3 Derivatives and Contiguous Functions
§16.3 Derivatives and Contiguous Functions
… ►§16.3(ii) Contiguous Functions
►Two generalized hypergeometric functions are (generalized) contiguous if they have the same pair of values of and , and corresponding parameters differ by integers. If , then any distinct contiguous functions are linearly related. Examples are provided by the following recurrence relations: …3: 17.6 Function
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Heine’s First Transformation
… ►Heine’s Second Tranformation
… ►Heine’s Third Transformation
… ►§17.6(iii) Contiguous Relations
►Heine’s Contiguous Relations
…4: 16.4 Argument Unity
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►The characterizing properties (18.22.2), (18.22.10), (18.22.19), (18.22.20), and (18.26.14) of the Hahn and Wilson class polynomials are examples of the contiguous relations mentioned in the previous three paragraphs.
►Contiguous balanced series have parameters shifted by an integer but still balanced.
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►See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations.
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5: 18.9 Recurrence Relations and Derivatives
§18.9 Recurrence Relations and Derivatives
►§18.9(i) Recurrence Relations
… ►with initial values and . … ► ►§18.9(ii) Contiguous Relations in the Parameters and the Degree
…6: 18.11 Relations to Other Functions
§18.11 Relations to Other Functions
… ►Ultraspherical
… ►Laguerre
… ►Hermite
… ►§18.11(ii) Formulas of Mehler–Heine Type
…7: Bibliography G
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Contiguous relations and summation and transformation formulae for basic hypergeometric series.
J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
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Some elementary inequalities relating to the gamma and incomplete gamma function.
J. Math. Phys. 38 (1), pp. 77–81.
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Computational aspects of three-term recurrence relations.
SIAM Rev. 9 (1), pp. 24–82.
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Questions of Numerical Condition Related to Polynomials.
In Studies in Numerical Analysis, G. H. Golub (Ed.),
pp. 140–177.
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Computation of Bessel and Airy functions and of related Gaussian quadrature formulae.
BIT 42 (1), pp. 110–118.
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8: 14.28 Sums
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§14.28(ii) Heine’s Formula
…9: Bibliography S
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A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations.
SIAM J. Math. Anal. 10 (4), pp. 823–838.
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Chebyshev expansions for the error and related functions.
Math. Comp. 32 (144), pp. 1232–1240.
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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros.
Math. Comp. 70 (235), pp. 1205–1220.
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Interlacing of the zeros of contiguous hypergeometric functions.
Numer. Algorithms 49 (1-4), pp. 387–407.
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Structure of avoided crossings for eigenvalues related to equations of Heun’s class.
J. Phys. A 30 (2), pp. 673–687.
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10: Bibliography V
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An integral transform involving Heun functions and a related eigenvalue problem.
SIAM J. Math. Anal. 17 (3), pp. 688–703.
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Expansions in products of Heine-Stieltjes polynomials.
Constr. Approx. 15 (4), pp. 467–480.
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Integral relations for Lamé functions.
SIAM J. Math. Anal. 13 (6), pp. 978–987.
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