contiguous relations (Heine)
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1: 15.5 Derivatives and Contiguous Functions
§15.5 Derivatives and Contiguous Functions
… โบ§15.5(ii) Contiguous Functions
โบThe six functions , , are said to be contiguous to . … โบAn 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
… โบ§16.3(ii) Contiguous Functions
โบTwo 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
… โบHeine’s Second Tranformation
… โบHeine’s Third Transformation
… โบ§17.6(iii) Contiguous Relations
โบHeine’s Contiguous Relations
…4: 16.4 Argument Unity
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โบThe 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.
โบContiguous balanced series have parameters shifted by an integer but still balanced.
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โบSee 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
โบ§18.9(i) Recurrence Relations
… โบwith initial values and . … โบ โบ§18.9(ii) Contiguous Relations in the Parameters and the Degree
…6: 18.11 Relations to Other Functions
§18.11 Relations to Other Functions
… โบUltraspherical
… โบLaguerre
… โบHermite
… โบ§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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