Relations for the Horn Functions
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
RELATIONS FOR THE HORN FUNCTIONS R. M. Mavlyaviev Kazan Federal University 18, Kremlevskaya St., Kazan 420008, Russia [email protected]
I. B. Garipov ∗ Kazan Federal University 18, Kremlevskaya St., Kazan 420008, Russia ilnur [email protected]
UDC 517.58
We prove Gauss type relations for the Horn functions H3 . Bibliography: 7 titles.
1
Introduction
Special functions that are solutions to ordinary differential equations or systems of such equations are widely used to solve problems in mathematical physics. For example, the Gaussian hypergeometric function ∞ (α)n (β)n z n F (α, β; δ; z) = (δ)n n! n=0
is a solution to the equation z(1 − z)uzz + (δ − (α + β + 1)z)uz − αβω = 0 and plays an important role in the theory of differential equations with the Bessel operator [1] uxx + uyy + uzz +
2α uz = 0. z
There are many relations connecting the function F (·) with different parameters. For example, the well-known 15 Gauss relations for contiguous hypergeometric functions (δ − 2α − (β − α)z)F (α, β; δ; z) + α(1 − z)F (α + 1, β; δ; z) − (δ − α)F (α − 1, β; δ; z) = 0,
∗
(β − α)F (α, β; δ; z) + αF (α + 1, β; δ; z) − βF (α, β + 1; δ; z) = 0,
(1.2)
(δ − α − β)F (α, β; δ; z) + α(1 − z)F (α + 1, β; δ; z) − (δ − β)F (α, β − 1; δ; z) = 0,
(1.3)
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 57-62. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0062
62
(1.1)
δ(α − (δ − β)z)F (α, β; δ; z) − αδ(1 − z)F (α + 1, β; δ; z) + (δ − α)(δ − β)zF (α, β; δ + 1; z) = 0,
(1.4)
(δ − α − 1)F (α, β; δ; z) + αF (α + 1, β; δ; z) − (δ − 1)F (α, β; δ − 1; z) = 0,
(1.5)
(δ − α − β)F (α, β; δ; z) − (δ − α)F (α − 1, β; δ; z) + β(1 − z)F (α, β + 1; δ; z) = 0,
(1.6)
(β − α)(1 − z)F (α, β; δ; z) − (δ − α)F (α − 1, β; δ; z) + (δ − β)F (α, β − 1; δ; z) = 0,
(1.7)
δ(1 − z)F (α, β; δ; z) − δF (α − 1, β; δ; z) + (δ − β)zF (α, β; δ + 1; z) = 0,
(1.8)
(α − 1 − (δ − β − 1)z)F (α, β; δ; z) + (δ − α)F (α − 1, β; δ; z) − (δ − 1)(1 − z)F (α, β; δ − 1; z) = 0,
(1.9)
(δ − 2β − (α − β)z)F (α, β; δ; z) + β(1 − z)F (α, β + 1; δ; z) − (δ − β)F (α, β − 1; δ; z) = 0,
(1.10)
δ(β − (δ − α)z)F (α, β; δ; z) − βδ(1 − z)F (α, β + 1; δ; z) + (δ − α)(δ − β)zF (α, β; δ + 1; z) = 0,
(1.11)
(δ − β − 1)F (α, β; δ; z) + βF (α, β + 1; δ; z) − (δ − 1)F (α, β; δ − 1; z) = 0,
(1.12)
δ(1 − z)F (α, β; δ; z) − δF (α, β − 1; δ; z) + (δ − α)zF (α, β; δ + 1; z) = 0,
(1.13)
(β − 1 − (δ − α − 1)z)F (α, β; δ; z) + (δ − β)F (α, β − 1; δ; z) − (δ − 1)(1 − z)F (α, β; δ − 1; z) = 0,
(1.14)
δ(δ − 1 − (2δ − α − β − 1)z)F (α, β; δ; z) + (δ − α)(δ − β)zF (α, β; δ + 1; z) − δ(δ − 1)(1 − z)F (α, β; δ − 1; z) = 0.
(1.15)
There are also relations connecting the hypergeometric function F (α, β; δ; z) with some F (α + l, β + m; δ + n; z), where l, m, n are arbitrary integers. For example, αβ zF (α + 1, β + 1; δ + 1; z), δ(1 − δ) α F (α, β + 1; δ; z) − F (α, β; δ; z) = zF (α + 1, β + 1; δ + 1; z). δ F (α, β; δ; z) − F (α, β; δ − 1; z)
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