# Series representations of Carlson symmetric elliptic integrals

Related topics: Carlson symmetric elliptic integrals

## Definitions

Symbol: CarlsonHypergeometricR $R_{-a}\!\left(b, z\right)$ Carlson multivariate hypergeometric function
Symbol: CarlsonHypergeometricT $T_{N}\!\left(b, z\right)$ Term in expansion of Carlson multivariate hypergeometric function

## Cases reducible to the Gauss hypergeometric function

Related topics: Gauss hypergeometric function

$R_C\!\left(1, x\right) = \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, 1 - x\right)$
$R_F\!\left(0, x, 1\right) = \frac{\pi}{2} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 1, 1 - x\right)$
$R_G\!\left(0, x, 1\right) = \frac{\pi}{4} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}, 1, 1 - x\right)$
$R_D\!\left(0, x, 1\right) = \frac{3 \pi}{4} \,{}_2F_1\!\left(\frac{1}{2}, \frac{3}{2}, 2, 1 - x\right)$
$R_D\!\left(0, 1, x\right) = \frac{3 \pi}{4 x} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}, 2, 1 - x\right)$

## Incomplete integrals

$R_F\!\left(x, y, z\right) = R_{-1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z\right]\right)$
$R_G\!\left(x, y, z\right) = R_{1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z\right]\right)$
$R_J\!\left(x, y, z, w\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, 1\right], \left[x, y, z, w\right]\right)$
$R_J\!\left(x, y, z, w\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z, w, w\right]\right)$
$R_D\!\left(x, y, z\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{3}{2}\right], \left[x, y, z\right]\right)$
$R_D\!\left(x, y, z\right) = R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, z, z, z\right]\right)$
$R_C\!\left(x, y\right) = R_{-1 / 2}\!\left(\left[\frac{1}{2}, 1\right], \left[x, y\right]\right)$
$R_C\!\left(x, y\right) = R_{-1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[x, y, y\right]\right)$

## Complete integrals

$R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[0, z_{2}, z_{3}, \ldots, z_{n}\right]\right) = \frac{\mathrm{B}\!\left(a, c - b_{1}\right)}{\mathrm{B}\!\left(a, c\right)} R_{-a}\!\left(\left[b_{2}, b_{3}, \ldots, b_{n}\right], \left[z_{2}, z_{3}, \ldots, z_{n}\right]\right)\; \text{ where } c = -a + \sum_{k=1}^{n} b_{k}$
$R_F\!\left(0, y, z\right) = \frac{\pi}{2} R_{-1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}\right], \left[y, z\right]\right)$
$R_D\!\left(0, y, z\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{3}{2}\right], \left[y, z\right]\right)$
$R_D\!\left(0, y, z\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[y, z, z, z\right]\right)$
$R_J\!\left(0, y, z, w\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, 1\right], \left[y, z, w\right]\right)$
$R_J\!\left(0, y, z, w\right) = \frac{3 \pi}{4} R_{-3 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right], \left[y, z, w, w\right]\right)$
$R_G\!\left(0, y, z\right) = \frac{\pi}{4} R_{1 / 2}\!\left(\left[\frac{1}{2}, \frac{1}{2}\right], \left[y, z\right]\right)$

## General formulas for the series

$T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{\textstyle{\left(m_{1}, m_{2}, \ldots, m_{n}\right) \in {\left(\mathbb{Z}_{\ge 0}\right)}^{n} \atop \sum_{k=1}^{n} m_{k} = N}} \prod_{k=1}^{n} \frac{\left(b_{k}\right)_{m_{k}}}{\left(m_{k}\right)!} z_{k}^{m_{k}}$
$R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[1 - z_{1}, 1 - z_{2}, \ldots, 1 - z_{n}\right]\right)\; \text{ where } c = \sum_{k=1}^{n} b_{k}$
$R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = z_{n}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(c\right)_{N}} T_{N}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n - 1}\right], \left[1 - \frac{z_{1}}{z_{n}}, 1 - \frac{z_{2}}{z_{n}}, \ldots, 1 - \frac{z_{n - 1}}{z_{n}}\right]\right)\; \text{ where } c = \sum_{k=1}^{n} b_{k}$

## Symmetric formulas

$R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[\frac{z_{1}}{A}, \frac{z_{2}}{A}, \ldots, \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}$
$R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = {A}^{-a} \sum_{N=0}^{\infty} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[1 - \frac{z_{1}}{A}, 1 - \frac{z_{2}}{A}, \ldots, 1 - \frac{z_{n}}{A}\right]\right)\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k}$
$T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) = \sum_{\textstyle{\left(m_{1}, m_{2}, \ldots, m_{n}\right) \in {\left(\mathbb{Z}_{\ge 0}\right)}^{n} \atop \sum_{k=1}^{n} k m_{k} = N}} {\left(-1\right)}^{M + N} \left(\beta\right)_{M} \prod_{k=1}^{n} \frac{e_{k}^{m_{k}}\!\left(\left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)}{\left(m_{k}\right)!}\; \text{ where } M = \sum_{k=1}^{n} m_{k}$
$R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\lambda z_{1}, \lambda z_{2}, \ldots, \lambda z_{n}\right]\right) = {\lambda}^{-a} R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)$
$R_{-a}\!\left(\left[b_{1}, b_{2}, \ldots, b_{n}\right], \left[\underbrace{z, \ldots, z}_{n \text{ times}}\right]\right) = {z}^{-a}$

## Approximations by truncated series

$\left|R_{-a}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right) - {A}^{-a} \sum_{N=0}^{K - 1} \frac{\left(a\right)_{N}}{\left(n \beta\right)_{N}} T_{N}\!\left(\left[\underbrace{\beta, \ldots, \beta}_{n \text{ times}}\right], \left[z_{1}, z_{2}, \ldots, z_{n}\right]\right)\right| \le \frac{\left|{A}^{-a}\right| \left(\left|a\right|\right)_{K} {M}^{K}}{K ! {\left(1 - M\right)}^{\max\left(\left|a\right|, 1\right)}}\; \text{ where } A = \frac{1}{n} \sum_{k=1}^{n} z_{k},\;Z_{k} = 1 - \frac{z_{k}}{A},\;M = \max\!\left(\left|Z_{1}\right|, \left|Z_{2}\right|, \ldots, \left|Z_{n}\right|\right)$
$\left|R_F\!\left(x, y, z\right) - {A}^{-1 / 2} \left(1 - \frac{E}{10} + \frac{F}{14} + \frac{{E}^{2}}{24} - \frac{3 E F}{44} - \frac{5 {E}^{3}}{208} + \frac{3 {F}^{2}}{104} + \frac{{E}^{2} F}{16}\right)\right| \le \frac{0.2 \left|{A}^{-1 / 2}\right| {M}^{8}}{1 - M}\; \text{ where } A = \frac{x + y + z}{3},\;X = 1 - \frac{x}{A},\;Y = 1 - \frac{y}{A},\;Z = 1 - \frac{z}{A},\;E = X Y + X Z + Y Z,\;F = X Y Z,\;M = \max\!\left(\left|X\right|, \left|Y\right|, \left|Z\right|\right)$
$\left|R_J\!\left(x, y, z, w\right) - {A}^{-3 / 2} \left(1 - \frac{3 E}{14} + \frac{F}{6} + \frac{9 {E}^{2}}{88} - \frac{3 G}{22} - \frac{9 E F}{52} + \frac{3 H}{26} - \frac{{E}^{3}}{16} + \frac{3 {F}^{2}}{40} + \frac{3 E G}{20} + \frac{45 {E}^{2} F}{272} - \frac{9 F G}{68} - \frac{9 E H}{68}\right)\right| \le \frac{3.4 \left|{A}^{-3 / 2}\right| {M}^{8}}{{\left(1 - M\right)}^{3 / 2}}\; \text{ where } A = \frac{x + y + z + 2 w}{5},\;X = 1 - \frac{x}{A},\;Y = 1 - \frac{y}{A},\;Z = 1 - \frac{z}{A},\;W = \frac{\left(-X - Y - Z\right)}{2},\;E = X Y + X Z + Y Z - 3 {W}^{2},\;F = X Y Z + 2 E W + 4 {W}^{3},\;G = \left(2 X Y Z + E W + 3 {W}^{3}\right) W,\;H = X Y Z {W}^{2},\;M = \max\!\left(\left|X\right|, \left|Y\right|, \left|Z\right|, \left|W\right|\right)$