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Gauss hypergeometric function

Table of contents: Hypergeometric series - Differential equations - Specific values - Symmetries - Linear fractional transformations - Bounds and inequalities

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Symbol: Hypergeometric2F1 2F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
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Symbol: Hypergeometric2F1Regularized 2F1 ⁣(a,b,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) Regularized Gauss hypergeometric function

Hypergeometric series

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2F1 ⁣(a,b,c,z)=k=0(a)k(b)k(c)kzkk!\,{}_2F_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}
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2F1 ⁣(a,b,c,z)=k=0(a)k(b)kΓ ⁣(c+k)zkk!\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\Gamma\!\left(c + k\right)} \frac{{z}^{k}}{k !}
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2F1 ⁣(a,b,c,z)=2F1 ⁣(a,b,c,z)Γ ⁣(c)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{\,{}_2F_1\!\left(a, b, c, z\right)}{\Gamma\!\left(c\right)}
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2F1 ⁣(a,b,n,z)=(a)n+1(b)n+1zn+1(n+1)!2F1 ⁣(a+n+1,b+n+1,n+2,z)\,{}_2{\textbf F}_1\!\left(a, b, -n, z\right) = \frac{\left(a\right)_{n + 1} \left(b\right)_{n + 1} {z}^{n + 1}}{\left(n + 1\right)!} \,{}_2F_1\!\left(a + n + 1, b + n + 1, n + 2, z\right)

Differential equations

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z(1z)y(z)+(c(a+b+1)z)y(z)aby ⁣(z)=0   where y ⁣(z)=2F1 ⁣(a,b,c,z)z \left(1 - z\right) y''(z) + \left(c - \left(a + b + 1\right) z\right) y'(z) - a b y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = \,{}_2F_1\!\left(a, b, c, z\right)

Specific values

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2F1 ⁣(a,b,c,0)=1\,{}_2F_1\!\left(a, b, c, 0\right) = 1
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2F1 ⁣(a,b,c,1)=Γ ⁣(c)Γ ⁣(cab)Γ ⁣(ca)Γ ⁣(cb)\,{}_2F_1\!\left(a, b, c, 1\right) = \frac{\Gamma\!\left(c\right) \Gamma\!\left(c - a - b\right)}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)}
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2F1 ⁣(1,1,2,z)=log ⁣(1z)z\,{}_2F_1\!\left(1, 1, 2, z\right) = -\frac{\log\!\left(1 - z\right)}{z}
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2F1 ⁣(a,b,b,z)=(1z)a\,{}_2F_1\!\left(a, b, b, z\right) = {\left(1 - z\right)}^{-a}

Symmetries

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2F1 ⁣(a,b,c,z)=2F1 ⁣(b,a,c,z)\,{}_2F_1\!\left(a, b, c, z\right) = \,{}_2F_1\!\left(b, a, c, z\right)
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2F1 ⁣(a,b,c,z)=2F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) = \overline{\,{}_2F_1\!\left(\overline{a}, \overline{b}, \overline{c}, \overline{z}\right)}

Linear fractional transformations

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2F1 ⁣(a,b,c,z)=(1z)cab2F1 ⁣(ca,cb,c,z)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{c - a - b} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c, z\right)
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2F1 ⁣(a,b,c,z)=(1z)a2F1 ⁣(a,cb,c,zz1)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-a} \,{}_2{\textbf F}_1\!\left(a, c - b, c, \frac{z}{z - 1}\right)
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2F1 ⁣(a,b,c,z)=(1z)b2F1 ⁣(ca,b,c,zz1)\,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = {\left(1 - z\right)}^{-b} \,{}_2{\textbf F}_1\!\left(c - a, b, c, \frac{z}{z - 1}\right)
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sin ⁣(π(ba))π2F1 ⁣(a,b,c,z)=(z)aΓ ⁣(b)Γ ⁣(ca)2F1 ⁣(a,ac+1,ab+1,1z)(z)bΓ ⁣(a)Γ ⁣(cb)2F1 ⁣(b,bc+1,ba+1,1z)\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b\right) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a - b + 1, \frac{1}{z}\right) - \frac{{\left(-z\right)}^{-b}}{\Gamma\!\left(a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, b - c + 1, b - a + 1, \frac{1}{z}\right)
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sin ⁣(π(ba))π2F1 ⁣(a,b,c,z)=(1z)aΓ ⁣(b)Γ ⁣(ca)2F1 ⁣(a,cb,ab+1,11z)(1z)bΓ ⁣(a)Γ ⁣(cb)2F1 ⁣(b,ca,ba+1,11z)\frac{\sin\!\left(\pi \left(b - a\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{\left(1 - z\right)}^{-a}}{\Gamma\!\left(b\right) \Gamma\!\left(c - a\right)} \,{}_2{\textbf F}_1\!\left(a, c - b, a - b + 1, \frac{1}{1 - z}\right) - \frac{{\left(1 - z\right)}^{-b}}{\Gamma\!\left(a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(b, c - a, b - a + 1, \frac{1}{1 - z}\right)
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sin ⁣(π(cab))π2F1 ⁣(a,b,c,z)=1Γ ⁣(ca)Γ ⁣(cb)2F1 ⁣(a,b,a+bc+1,1z)(1z)cabΓ ⁣(a)Γ ⁣(b)2F1 ⁣(ca,cb,cab+1,1z)\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{1}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, b, a + b - c + 1, 1 - z\right) - \frac{{\left(1 - z\right)}^{c - a - b}}{\Gamma\!\left(a\right) \Gamma\!\left(b\right)} \,{}_2{\textbf F}_1\!\left(c - a, c - b, c - a - b + 1, 1 - z\right)
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sin ⁣(π(cab))π2F1 ⁣(a,b,c,z)=zaΓ ⁣(ca)Γ ⁣(cb)2F1 ⁣(a,ac+1,a+bc+1,11z)zac(1z)cabΓ ⁣(a)Γ ⁣(b)2F1 ⁣(ca,1a,cab+1,11z)\frac{\sin\!\left(\pi \left(c - a - b\right)\right)}{\pi} \,{}_2{\textbf F}_1\!\left(a, b, c, z\right) = \frac{{z}^{-a}}{\Gamma\!\left(c - a\right) \Gamma\!\left(c - b\right)} \,{}_2{\textbf F}_1\!\left(a, a - c + 1, a + b - c + 1, 1 - \frac{1}{z}\right) - \frac{{z}^{a - c} {\left(1 - z\right)}^{c - a - b}}{\Gamma\!\left(a\right) \Gamma\!\left(b\right)} \,{}_2{\textbf F}_1\!\left(c - a, 1 - a, c - a - b + 1, 1 - \frac{1}{z}\right)

Bounds and inequalities

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2F1 ⁣(a,b,c,z)k=0N1(a)k(b)k(c)kzkk!(a)N(b)N(c)NzNN!11D   where D=z(1+acc+N)(1+b11+N)\left|\,{}_2F_1\!\left(a, b, c, z\right) - \sum_{k=0}^{N - 1} \frac{\left(a\right)_{k} \left(b\right)_{k}}{\left(c\right)_{k}} \frac{{z}^{k}}{k !}\right| \le \left|\frac{\left(a\right)_{N} \left(b\right)_{N}}{\left(c\right)_{N}} \frac{{z}^{N}}{N !}\right| \frac{1}{1 - D}\; \text{ where } D = \left|z\right| \left(1 + \frac{\left|a - c\right|}{\left|c + N\right|}\right) \left(1 + \frac{\left|b - 1\right|}{\left|1 + N\right|}\right)
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f(k)(z)k!A(N+kk)νk   where f ⁣(z)=2F1 ⁣(a,b,c,z),ν=max ⁣(1z1,1z),N=2max ⁣(ν1ab,a+b+1+2c),A=max ⁣(f ⁣(z),f(z)ν(N+1))\left|\frac{{f}^{(k)}(z)}{k !}\right| \le A {N + k \choose k} {\nu}^{k}\; \text{ where } f\!\left(z\right) = \,{}_2{\textbf F}_1\!\left(a, b, c, z\right),\,\nu = \max\!\left(\frac{1}{\left|z - 1\right|}, \frac{1}{\left|z\right|}\right),\,N = 2 \max\!\left(\sqrt{{\nu}^{-1} \left|a b\right|}, \left|a + b + 1\right| + 2 \left|c\right|\right),\,A = \max\!\left(\left|f\!\left(z\right)\right|, \frac{\left|f'(z)\right|}{\nu \left(N + 1\right)}\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-09-19 20:12:49.583742 UTC