Fungrim home page

Confluent hypergeometric functions

Table of contents: Hypergeometric series - Differential equations - Kummer's transformation - Connection formulas - Asymptotic expansions

316533
Symbol: Hypergeometric0F1 0F1 ⁣(a,z)\,{}_0F_1\!\left(a, z\right) Confluent hypergeometric limit function
f565f5
Symbol: Hypergeometric0F1Regularized 0F1 ⁣(a,z)\,{}_0{\textbf F}_1\!\left(a, z\right) Regularized confluent hypergeometric limit function
512bea
Symbol: Hypergeometric1F1 1F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric function
cee331
Symbol: Hypergeometric1F1Regularized 1F1 ⁣(a,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) Regularized Kummer confluent hypergeometric function
d6add6
Symbol: HypergeometricU U ⁣(a,b,z)U\!\left(a, b, z\right) Tricomi confluent hypergeometric function
1b9cc5
Symbol: HypergeometricUStar U ⁣(a,b,z)U^{*}\!\left(a, b, z\right) Scaled Tricomi confluent hypergeometric function
b9cc75
Symbol: Hypergeometric2F0 2F0 ⁣(a,b,z)\,{}_2F_0\!\left(a, b, z\right) Tricomi confluent hypergeometric function, alternative notation

Hypergeometric series

4c41ad
0F1 ⁣(a,z)=k=01(a)kzkk!\,{}_0F_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\left(a\right)_{k}} \frac{{z}^{k}}{k !}
0a0aec
0F1 ⁣(a,z)=k=01Γ ⁣(a+k)zkk!\,{}_0{\textbf F}_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\Gamma\!\left(a + k\right)} \frac{{z}^{k}}{k !}
a61f01
1F1 ⁣(a,b,z)=k=0(a)k(b)kzkk!\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}
dec042
1F1 ⁣(n,b,z)=k=0n(n)k(b)kzkk!\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}
70111e
1F1 ⁣(a,b,z)=k=0(a)kΓ ⁣(b+k)zkk!\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\Gamma\!\left(b + k\right)} \frac{{z}^{k}}{k !}

Differential equations

06f229
zy(z)+(bz)y(z)ay(z)=0   where y(z)=C1F1 ⁣(a,b,z)+DU ⁣(a,b,z)z y''(z) + \left(b - z\right) y'(z) - a y(z) = 0\; \text{ where } y(z) = C \,{}_1{\textbf F}_1\!\left(a, b, z\right) + D U\!\left(a, b, z\right)
bb5d67
zy(z)+ay(z)y(z)=0   where y(z)=C0F1 ⁣(a,z)+Dz1a0F1 ⁣(2a,z)z y''(z) + a y'(z) - y(z) = 0\; \text{ where } y(z) = C \,{}_0{\textbf F}_1\!\left(a, z\right) + D {z}^{1 - a} \,{}_0{\textbf F}_1\!\left(2 - a, z\right)

Kummer's transformation

be533c
1F1 ⁣(a,b,z)=ez1F1 ⁣(ba,b,z)\,{}_1F_1\!\left(a, b, z\right) = {e}^{z} \,{}_1F_1\!\left(b - a, b, -z\right)
a047eb
1F1 ⁣(a,b,z)=ez1F1 ⁣(ba,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) = {e}^{z} \,{}_1{\textbf F}_1\!\left(b - a, b, -z\right)
9d3147
U ⁣(a,b,z)=z1bU ⁣(1+ab,2b,z)U\!\left(a, b, z\right) = {z}^{1 - b} U\!\left(1 + a - b, 2 - b, z\right)

Connection formulas

c8fcc7
U ⁣(a,b,z)=zaU ⁣(a,b,z)U^{*}\!\left(a, b, z\right) = {z}^{a} U\!\left(a, b, z\right)
4cf1e9
U ⁣(a,b,z)=2F0 ⁣(a,ab+1,1z)U^{*}\!\left(a, b, z\right) = \,{}_2F_0\!\left(a, a - b + 1, -\frac{1}{z}\right)
f7f84e
1F1 ⁣(a,b,z)=(z)aΓ ⁣(ba)U ⁣(a,b,z)+zabezΓ(a)U ⁣(ba,b,z)\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma(a)} U^{*}\!\left(b - a, b, -z\right)
6cf802
U ⁣(a,b,z)=Γ ⁣(1b)Γ ⁣(ab+1)1F1 ⁣(a,b,z)+Γ ⁣(b1)Γ(a)z1b1F1 ⁣(ab+1,2b,z)U\!\left(a, b, z\right) = \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma(a)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)
18ef23
U ⁣(a,n,z)=limbnΓ ⁣(1b)Γ ⁣(ab+1)1F1 ⁣(a,b,z)+Γ ⁣(b1)Γ(a)z1b1F1 ⁣(ab+1,2b,z)U\!\left(a, n, z\right) = \lim_{b \to n} \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma(a)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)
2df3e3
0F1 ⁣(a,z)=e2z1F1 ⁣(a12,2a1,4z)\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)
325a0e
0F1 ⁣(a,z)=(z)(1a)/2Ja1 ⁣(2z)\,{}_0{\textbf F}_1\!\left(a, z\right) = {\left(-z\right)}^{\left( 1 - a \right) / 2} J_{a - 1}\!\left(2 \sqrt{-z}\right)
00dfd1
0F1 ⁣(a,z)=z(1a)/2Ia1 ⁣(2z)\,{}_0{\textbf F}_1\!\left(a, z\right) = {z}^{\left( 1 - a \right) / 2} I_{a - 1}\!\left(2 \sqrt{z}\right)

Asymptotic expansions

d1b3b5
U ⁣(a,b,z)=k=0n1(a)k(ab+1)kk!(z)k+Rn ⁣(a,b,z)U^{*}\!\left(a, b, z\right) = \sum_{k=0}^{n - 1} \frac{\left(a\right)_{k} \left(a - b + 1\right)_{k}}{k ! {\left(-z\right)}^{k}} + R_{n}\!\left(a,b,z\right)
99f69c
Symbol: HypergeometricUStarRemainder Rn ⁣(a,b,z)R_{n}\!\left(a,b,z\right) Error term in asymptotic expansion of Tricomi confluent hypergeometric function
876844
limzRn ⁣(a,b,eiθz)=0\lim_{z \to \infty} \left|R_{n}\!\left(a,b,{e}^{i \theta} z\right)\right| = 0
279e4f
Rn ⁣(a,b,z)(a)n(ab+1)nn!zn21σexp ⁣(2ρ(1σ)z)   where σ=b2az,ρ=a2ab+b2+σ(1+σ4)(1σ)2\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2}{1 - \sigma} \exp\!\left(\frac{2 \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}
461a54
Rn ⁣(a,b,z)(a)n(ab+1)nn!zn21+12πn1σexp ⁣(πρ(1σ)z)   where σ=b2az,ρ=a2ab+b2+σ(1+σ4)(1σ)2\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 \sqrt{1 + \frac{1}{2} \pi n}}{1 - \sigma} \exp\!\left(\frac{\pi \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}
7b91b4
Rn ⁣(a,b,z)(a)n(ab+1)nn!zn2C(n)1τexp ⁣(2C(1)ρ(1τ)z)   where σ=b2az,ν=1+2σ2,τ=νσ,ρ=a2ab+b2+τ(1+τ4)(1τ)2,C(m)=(1+πm2+σν2m)νm\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 C(n)}{1 - \tau} \exp\!\left(\frac{2 C(1) \rho}{\left(1 - \tau\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\nu = 1 + 2 {\sigma}^{2},\,\tau = \nu \sigma,\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\tau \left(1 + \frac{\tau}{4}\right)}{{\left(1 - \tau\right)}^{2}},\,C(m) = \left(\sqrt{1 + \frac{\pi m}{2}} + \sigma {\nu}^{2} m\right) {\nu}^{m}

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

2019-11-11 15:50:15.016492 UTC