# Confluent hypergeometric functions

Symbol: Hypergeometric0F1 $\,{}_0F_1\!\left(a, z\right)$ Confluent hypergeometric limit function
Symbol: Hypergeometric0F1Regularized $\,{}_0{\textbf F}_1\!\left(a, z\right)$ Regularized confluent hypergeometric limit function
Symbol: Hypergeometric1F1 $\,{}_1F_1\!\left(a, b, z\right)$ Kummer confluent hypergeometric function
Symbol: Hypergeometric1F1Regularized $\,{}_1{\textbf F}_1\!\left(a, b, z\right)$ Regularized Kummer confluent hypergeometric function
Symbol: HypergeometricU $U\!\left(a, b, z\right)$ Tricomi confluent hypergeometric function
Symbol: HypergeometricUStar $U^{*}\!\left(a, b, z\right)$ Scaled Tricomi confluent hypergeometric function
Symbol: Hypergeometric2F0 $\,{}_2F_0\!\left(a, b, z\right)$ Tricomi confluent hypergeometric function, alternative notation

## Hypergeometric series

$\,{}_0F_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\left(a\right)_{k}} \frac{{z}^{k}}{k !}$
$\,{}_0{\textbf F}_1\!\left(a, z\right) = \sum_{k=0}^{\infty} \frac{1}{\Gamma\!\left(a + k\right)} \frac{{z}^{k}}{k !}$
$\,{}_1F_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}$
$\,{}_1F_1\!\left(-n, b, z\right) = \sum_{k=0}^{n} \frac{\left(-n\right)_{k}}{\left(b\right)_{k}} \frac{{z}^{k}}{k !}$
$\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \sum_{k=0}^{\infty} \frac{\left(a\right)_{k}}{\Gamma\!\left(b + k\right)} \frac{{z}^{k}}{k !}$

## Differential equations

$z y''(z) + \left(b - z\right) y'(z) - a y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \,{}_1{\textbf F}_1\!\left(a, b, z\right) + D U\!\left(a, b, z\right)$
$z y''(z) + a y'(z) - y\!\left(z\right) = 0\; \text{ where } y\!\left(z\right) = C \,{}_0{\textbf F}_1\!\left(a, z\right) + D {z}^{1 - a} \,{}_0{\textbf F}_1\!\left(2 - a, z\right)$

## Kummer's transformation

$\,{}_1F_1\!\left(a, b, z\right) = {e}^{z} \,{}_1F_1\!\left(b - a, b, -z\right)$
$\,{}_1{\textbf F}_1\!\left(a, b, z\right) = {e}^{z} \,{}_1{\textbf F}_1\!\left(b - a, b, -z\right)$
$U\!\left(a, b, z\right) = {z}^{1 - b} U\!\left(1 + a - b, 2 - b, z\right)$

## Connection formulas

$U^{*}\!\left(a, b, z\right) = {z}^{a} U\!\left(a, b, z\right)$
$U^{*}\!\left(a, b, z\right) = \,{}_2F_0\!\left(a, a - b + 1, -\frac{1}{z}\right)$
$\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma\!\left(a\right)} U^{*}\!\left(b - a, b, -z\right)$
$U\!\left(a, b, z\right) = \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma\!\left(a\right)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)$
$U\!\left(a, n, z\right) = \lim_{b \to n} \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma\!\left(a\right)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)$
$\,{}_0F_1\!\left(a, z\right) = {e}^{-2 \sqrt{z}} \,{}_1F_1\!\left(a - \frac{1}{2}, 2 a - 1, 4 \sqrt{z}\right)$
$\,{}_0{\textbf F}_1\!\left(a, z\right) = {\left(-z\right)}^{\left( 1 - a \right) / 2} J_{a - 1}\!\left(2 \sqrt{-z}\right)$
$\,{}_0{\textbf F}_1\!\left(a, z\right) = {z}^{\left( 1 - a \right) / 2} I_{a - 1}\!\left(2 \sqrt{z}\right)$

## Asymptotic expansions

$U^{*}\!\left(a, b, z\right) = \sum_{k=0}^{n - 1} \frac{\left(a\right)_{k} \left(a - b + 1\right)_{k}}{k ! {\left(-z\right)}^{k}} + R_{n}\!\left(a,b,z\right)$
Symbol: HypergeometricUStarRemainder $R_{n}\!\left(a,b,z\right)$ Error term in asymptotic expansion of Tricomi confluent hypergeometric function
$\lim_{z \to \infty} \left|R_{n}\!\left(a,b,{e}^{i \theta} z\right)\right| = 0$
$\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2}{1 - \sigma} \exp\!\left(\frac{2 \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}$
$\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 \sqrt{1 + \frac{1}{2} \pi n}}{1 - \sigma} \exp\!\left(\frac{\pi \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}$
$\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 C\!\left(n\right)}{1 - \tau} \exp\!\left(\frac{2 C\!\left(1\right) \rho}{\left(1 - \tau\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\,\nu = 1 + 2 {\sigma}^{2},\,\tau = \nu \sigma,\,\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\tau \left(1 + \frac{\tau}{4}\right)}{{\left(1 - \tau\right)}^{2}},\,C\!\left(m\right) = \left(\sqrt{1 + \frac{\pi m}{2}} + \sigma {\nu}^{2} m\right) {\nu}^{m}$

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

2019-09-20 18:07:53.062439 UTC