# Gamma and related functions Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **162 identities** for gamma and related functions. :::info[Generated reference] This page is generated from the compiled Fungrim artifact by `scripts/fungrim/gen-reference-doc.ts` (upstream snapshot `3a299164c683`, translator `grim2mathjson 0.1.0`). Do not edit it by hand. The corpus is MIT-licensed; see `data/fungrim/LICENSE`. ::: ## Contents - [Barnes G-function](#barnes-g-function) (27) - [Beta function](#beta-function) (13) - [Digamma function](#digamma-function) (66) - [Factorials and binomial coefficients](#factorials-and-binomial-coefficients) (32) - [Gamma function](#gamma-function) (24) ## Barnes G-function $$\mathrm{LogBarnesG}(1+z)=\frac{1}{2}((\ln(2\pi)-1)z)-\frac{1}{2}((1+\gamma)z^2)+\sum_{n=3}^{\infty}\frac{1}{n}(\Zeta(n-1)\times(-1)^{n+1}z^{n})$$ **Holds when** $z\in\C\land\vert z\vert\lt1$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`0ad263` · Fungrim entry ↗](https://fungrim.org/entry/0ad263) --- $$\mathrm{BarnesG}(z^\star)=\mathrm{BarnesG}(z)^\star$$ **Holds when** $z\in\C$. **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for expansion. [`147db6` · Fungrim entry ↗](https://fungrim.org/entry/147db6) --- $$\mathrm{LogBarnesG}(1-z)=\mathrm{LogBarnesG}(1+z)-\ln(2\pi)z+\begin{cases}\int_{0}^{\imaginaryI}\!\pi x\cot(\pi x)\, \mathrm{d}x+\int_{\imaginaryI}^{z}\!\pi x\cot(\pi x)\, \mathrm{d}x&-1\lt\Re(z)\lt1\lor\Im(z)\gt0\lor\Im(z)=0\land\Re(z)\lt1\\\int_{0}^{-\imaginaryI}\!\pi x\cot(\pi x)\, \mathrm{d}x+\int_{-\imaginaryI}^{z}\!\pi x\cot(\pi x)\, \mathrm{d}x&-1\lt\Re(z)\lt1\lor\Im(z)\lt0\lor\Im(z)=0\land\Re(z)\gt-1\end{cases}$$ **Holds when** $z\in\C\land z\notin\Z$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`23ed69` · Fungrim entry ↗](https://fungrim.org/entry/23ed69) --- $$\mathrm{BarnesG}(n)=\begin{cases}\prod_{k=1}^{n-2}k!&n\ge1\\0&n\le0\end{cases}$$ **Holds when** $n\in\Z$. **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. [`33f13a` · Fungrim entry ↗](https://fungrim.org/entry/33f13a) --- $$\mathrm{LogBarnesG}(z+1)=\mathrm{GammaLn}(z)+\mathrm{LogBarnesG}(z)$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`5261e3` · Fungrim entry ↗](https://fungrim.org/entry/5261e3) --- $$\mathrm{BarnesG}(1-x)=(-1)^{\lfloor\frac{x-1}{2}\rfloor+1}\mathrm{BarnesG}(1+x)\frac{\vert\sin(\pi x)\vert}{\pi}^{x}\exp(\frac{\Im(\operatorname{Li}_{2}(\exp(2\imaginaryI\pi x)))}{2\pi})$$ **Holds when** $x\in\R\land x\notin-\infty..-1$. **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. **Reference:** [doi.org](https://doi.org/10.1145/384101.384104) [`541e2e` · Fungrim entry ↗](https://fungrim.org/entry/541e2e) --- $$\mathrm{LogBarnesG}(x)=\begin{cases}\ln(\mathrm{BarnesG}(x))&x\gt0\\\ln(\vert\mathrm{BarnesG}(x)\vert)+\frac{\lfloor x\rfloor}{2}(\lfloor x\rfloor-1)\pi\imaginaryI&\top\end{cases}$$ **Holds when** $x\in\R\land x\notin\Z_{\le0}$. **Symbols:** **BarnesG** — Barnes G-function; **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`5a11eb` · Fungrim entry ↗](https://fungrim.org/entry/5a11eb) --- $$z\mapsto\mathrm{BarnesG}(z)^{\prime}(z)=\mathrm{BarnesG}(z)((z-1)\mathrm{Digamma}(z)-z+\frac{1}{2}(\ln(2\pi)+1))$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. [`5babc2` · Fungrim entry ↗](https://fungrim.org/entry/5babc2) --- $$\mathrm{LogBarnesG}(z+1)=(z\mathrm{GammaLn}(z)+z^2/4)-(\ln(z)\mathrm{BernoulliPolynomial}(2, z))/2-\ln(\mathrm{ConstGlaisher})-\int_{0}^{\infty}\!(((-x)/12-1/x+1/(1-\exp(-x))-1/2)\exp(-(xz)))/x^2\, \mathrm{d}x$$ **Holds when** $z\in\C\land\Re(z)\gt0$. **Symbols:** **BernoulliPolynomial** — Bernoulli polynomial; **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. **Reference:** [arxiv.org](https://arxiv.org/abs/math/0308086) [`6395ee` · Fungrim entry ↗](https://fungrim.org/entry/6395ee) --- $$\mathrm{LogBarnesG}(z^\star)=\begin{cases}\mathrm{LogBarnesG}(z)&z\in\lparen-\infty, 0\rbrack\\\mathrm{LogBarnesG}(z)^\star&\top\end{cases}$$ **Holds when** $z\in\C$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`6c6d3e` · Fungrim entry ↗](https://fungrim.org/entry/6c6d3e) --- $$\mathrm{LogBarnesG}(z+1)=(\frac{z^2}{4}+z\mathrm{GammaLn}(z+1))-(\frac{z(z+1)}{2}+\frac{1}{12})\ln(z)-\ln(\mathrm{ConstGlaisher})+\sum_{n=1}^{N_{var}-1}\frac{\mathrm{BernoulliB}(2n+2)}{2n(2n+1)(2n+2)z^{2n}}+\mathrm{LogBarnesGRemainder}(N_{var}, z)$$ **Holds when** $z\in\C\land z\notin\lparen-\infty, 0\rbrack\land N_{var}\in\N^*$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function; **LogBarnesGRemainder** — Remainder term in asymptotic expansion of logarithmic Barnes G-function. Used by the Compute Engine for simplification. **Reference:** [dx.doi.org](https://dx.doi.org/10.1098/rspa.2014.0534) [`6f8e14` · Fungrim entry ↗](https://fungrim.org/entry/6f8e14) --- $$\mathrm{LogBarnesG}(1-z)=\mathrm{LogBarnesG}(1+z)+\begin{cases}(\pi\imaginaryI(z^2-z+1/6))/2-z(\mathrm{GammaLn}(z)+\mathrm{GammaLn}(1-z))-\frac{1}{\pi}((1/2\imaginaryI)\operatorname{Li}_{2}(\exp(2\imaginaryI\pi z)))&0\lt\Re(z)\lt1\lor\Im(z)\gt0\lor\Im(z)=0\land\Re(z)\lt1\\-((\pi\imaginaryI((-z)^2-(-z)+1/6))/2-(-z(\mathrm{GammaLn}(-z)+\mathrm{GammaLn}(1-(-z))))-((1/2\imaginaryI)\operatorname{Li}_{2}(\exp(-2\imaginaryI\pi z)))/\pi)&-1\lt\Re(z)\lt0\lor\Im(z)\lt0\lor\Im(z)=0\land\Re(z)\gt-1\end{cases}$$ **Holds when** $z\in\C\land z\notin\Z$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`82b410` · Fungrim entry ↗](https://fungrim.org/entry/82b410) --- $$\mathrm{BarnesG}(z+1)=\Gamma(z)\mathrm{BarnesG}(z)$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. [`86b3ec` · Fungrim entry ↗](https://fungrim.org/entry/86b3ec) --- $$\mathrm{BarnesG}(\frac{1}{2})=\frac{\sqrt[24]{2}\exp(\frac{1}{8})}{\sqrt[4]{\pi}\sqrt{\mathrm{ConstGlaisher}}^{3}}$$ **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. [`8b7991` · Fungrim entry ↗](https://fungrim.org/entry/8b7991) --- $$\mathrm{LogBarnesG}(z+1)=(\frac{z(1-z)}{2}+\frac{1}{2}(z\ln(2\pi))+z\mathrm{GammaLn}(z))-\int_{0}^{z}\!\mathrm{GammaLn}(x)\, \mathrm{d}x$$ **Holds when** $z\in\C\land z\notin\lparen-\infty, -1\rbrack$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. **Reference:** [arxiv.org](https://arxiv.org/abs/math/0308086) [`8c96a5` · Fungrim entry ↗](https://fungrim.org/entry/8c96a5) --- $$\mathrm{LogBarnesG}(z+1)=\frac{z(1-z)}{2}+\frac{1}{2}(z\ln(2\pi))+\int_{0}^{z}\!x\mathrm{Digamma}(x)\, \mathrm{d}x$$ **Holds when** $z\in\C\land z\notin\lparen-\infty, -1\rbrack$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. **Reference:** [arxiv.org](https://arxiv.org/abs/math/0308086) [`95f771` · Fungrim entry ↗](https://fungrim.org/entry/95f771) --- $$\Im(\mathrm{LogBarnesG}(x))=\frac{1}{2}(\lfloor x\rfloor(\lfloor x\rfloor-1)\pi)$$ **Holds when** $x\in\R\land x\lt0\land x\notin\Z$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`a044e1` · Fungrim entry ↗](https://fungrim.org/entry/a044e1) --- $$z\mapsto\mathrm{LogBarnesG}(z)^{\prime}(z)=(z-1)\mathrm{Digamma}(z)-z+\frac{1}{2}(\ln(2\pi)+1)$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`af31ae` · Fungrim entry ↗](https://fungrim.org/entry/af31ae) --- $$\mathrm{BarnesG}(z)=\exp(\mathrm{LogBarnesG}(z))$$ **Holds when** $z\in\C$. **Symbols:** **BarnesG** — Barnes G-function; **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for expansion. [`b4355e` · Fungrim entry ↗](https://fungrim.org/entry/b4355e) --- $$\mathrm{LogBarnesG}(1-z)=\mathrm{LogBarnesG}(1+z)-\ln(2\pi)z+\int_{0}^{z}\!\pi x\cot(\pi x)\, \mathrm{d}x$$ **Holds when** $z\in\C\land z\notin\lparen-\infty, -1\rbrack\cup\lbrack1, \infty\rparen$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`b6017f` · Fungrim entry ↗](https://fungrim.org/entry/b6017f) --- $$\mathrm{LogBarnesG}(z+1)=(\frac{1}{4}(z^2(2\ln(z)-3))+\frac{1}{2}(z\ln(2\pi))+\frac{1}{12})-\ln(\mathrm{ConstGlaisher})-\int_{0}^{\infty}\!\frac{x\ln(x^2+z^2)}{\exp(2\pi x)-1}\, \mathrm{d}x$$ **Holds when** $z\in\C\land\Re(z)\gt0$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. **Reference:** [arxiv.org](https://arxiv.org/abs/math/0308086) [`b64782` · Fungrim entry ↗](https://fungrim.org/entry/b64782) --- $$\mathrm{BarnesG}(\frac{1}{4})=\frac{\exp(3/32-G/(4\pi))}{\mathrm{ConstGlaisher}^{\frac{9}{8}}\Gamma(1/4)^{\frac{3}{4}}}$$ **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. [`ce66a9` · Fungrim entry ↗](https://fungrim.org/entry/ce66a9) --- $$\mathrm{LogBarnesG}(1-x)=\mathrm{LogBarnesG}(1+x)+x\ln(\frac{\vert\sin(\pi x)\vert}{\pi})+\frac{\Im(\operatorname{Li}_{2}(\exp(2\imaginaryI\pi x)))}{2\pi}+\frac{1}{2}(\mathrm{sgn}(x)\lfloor x\rfloor(\lfloor x\rfloor+1)\pi\imaginaryI)$$ **Holds when** $x\in\R\land x\notin\Z$. **Symbols:** **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`d1a0ec` · Fungrim entry ↗](https://fungrim.org/entry/d1a0ec) --- $$\mathrm{LogBarnesG}(n)=\begin{cases}\ln(\mathrm{BarnesG}(n))&n\ge1\\-\infty&n\le0\end{cases}$$ **Holds when** $n\in\Z$. **Symbols:** **BarnesG** — Barnes G-function; **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`daef08` · Fungrim entry ↗](https://fungrim.org/entry/daef08) --- $$\mathrm{BarnesG}(\frac{3}{4})=\frac{\exp(\frac{3}{32}+\frac{G}{4\pi})\sqrt[4]{\Gamma(\frac{1}{4})}}{\sqrt[8]{2}\sqrt[4]{\pi}\mathrm{ConstGlaisher}^{\frac{9}{8}}}$$ **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. [`dc507f` · Fungrim entry ↗](https://fungrim.org/entry/dc507f) --- $$\mathrm{LogBarnesG}(z)=(z-1)\mathrm{GammaLn}(z)-\mathrm{HurwitzZeta}(-1, z, 1)+s\mapsto\Zeta(s)^{\prime}(-1)$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function; **LogBarnesG** — Logarithmic Barnes G-function. Used by the Compute Engine for simplification. [`e05807` · Fungrim entry ↗](https://fungrim.org/entry/e05807) --- $$z\mapsto\mathrm{BarnesG}(z)^{\prime}(n)=\begin{cases}0&n\lt0\\1&n=0\\\frac{1}{2}(\ln(2\pi)-1)&n=1\\\mathrm{BarnesG}(n)(\frac{\ln(2\pi)}{2}+(n-1)(\mathrm{HarmonicNumber}(n-2)-\gamma-1)+\frac{1}{2})&n\ge2\end{cases}$$ **Holds when** $n\in\Z$. **Symbols:** **BarnesG** — Barnes G-function. Used by the Compute Engine for simplification. [`f50c74` · Fungrim entry ↗](https://fungrim.org/entry/f50c74) --- ## Beta function $$\Beta(m, n)=\frac{(m-1)!(n-1)!}{((m+n)-1)!}$$ **Holds when** $m\in\N^*\land n\in\N^*$. Used by the Compute Engine for simplification. [`082a69` · Fungrim entry ↗](https://fungrim.org/entry/082a69) --- $$\mathrm{IncompleteBeta}(1, a, b)=\Beta(a, b)$$ **Holds when** $a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}$. **Symbols:** **IncompleteBeta** — Incomplete beta function. Used by the Compute Engine for expansion. [`3141e4` · Fungrim entry ↗](https://fungrim.org/entry/3141e4) --- $$\mathrm{IncompleteBetaRegularized}(x, a, b)=1-\mathrm{IncompleteBetaRegularized}(1-x, b, a)$$ **Holds when** $x\in\C\land a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}\land a+b\notin\Z_{\le0}$. **Symbols:** **IncompleteBetaRegularized** — Regularized incomplete beta function. Used by the Compute Engine for simplification. [`315b3d` · Fungrim entry ↗](https://fungrim.org/entry/315b3d) --- $$\Beta(n, b)=\begin{cases}\tilde\infty&-b\in0..n-1\\(n\binom{(n+b)-1}{n})^{-1}&\top\end{cases}$$ **Holds when** $n\in\N^*\land b\in\C$. Used by the Compute Engine for simplification. [`72db94` · Fungrim entry ↗](https://fungrim.org/entry/72db94) --- $$\Beta(a, b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$ **Holds when** $a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for simplification. [`888581` · Fungrim entry ↗](https://fungrim.org/entry/888581) --- $$\Beta(-n, b)=\begin{cases}\frac{(-1)^{b}}{b\binom{n}{b}}&b\in1..n\\\tilde\infty&\top\end{cases}$$ **Holds when** $n\in\N\land b\in\C$. Used by the Compute Engine for simplification. [`a7dbf6` · Fungrim entry ↗](https://fungrim.org/entry/a7dbf6) --- $$\mathrm{IncompleteBeta}(0, a, b)=0$$ **Holds when** $a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}$. **Symbols:** **IncompleteBeta** — Incomplete beta function. Used by the Compute Engine for simplification. [`ba7baf` · Fungrim entry ↗](https://fungrim.org/entry/ba7baf) --- $$\Beta(m, n)=(m\binom{(m+n)-1}{m})^{-1}$$ **Holds when** $m\in\N^*\land n\in\N^*$. Used by the Compute Engine for simplification. [`bb4f41` · Fungrim entry ↗](https://fungrim.org/entry/bb4f41) --- $$(a+b)\Beta(a+1, b)=a\Beta(a, b)$$ **Holds when** $a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for simplification. [`bdea17` · Fungrim entry ↗](https://fungrim.org/entry/bdea17) --- $$\mathrm{IncompleteBetaRegularized}(x, a, b)=\frac{\mathrm{IncompleteBeta}(x, a, b)}{\Beta(a, b)}$$ **Holds when** $x\in\C\land a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}\land a+b\notin\Z_{\le0}$. **Symbols:** **IncompleteBeta** — Incomplete beta function; **IncompleteBetaRegularized** — Regularized incomplete beta function. Used by the Compute Engine for simplification. [`c92da4` · Fungrim entry ↗](https://fungrim.org/entry/c92da4) --- $$\Beta(a, b)=\Beta(b, a)$$ **Holds when** $a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for expansion. [`cc2ebb` · Fungrim entry ↗](https://fungrim.org/entry/cc2ebb) --- $$\Beta(a, b)=\Beta(a+1, b)+\Beta(a, b+1)$$ **Holds when** $a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for simplification. [`e9f966` · Fungrim entry ↗](https://fungrim.org/entry/e9f966) --- $$\Beta(a, b)\Beta(a+b, c)=\Beta(b, c)\Beta(a, b+c)$$ **Holds when** $a\in\C\setminus\Z_{\le0}\land b\in\C\setminus\Z_{\le0}\land c\in\C\setminus\Z_{\le0}\land a+b\notin\Z_{\le0}\land b+c\notin\Z_{\le0}$. Used by the Compute Engine for expansion. [`fd0e48` · Fungrim entry ↗](https://fungrim.org/entry/fd0e48) --- ## Digamma function $$\mathrm{Digamma}(n)=\mathrm{HarmonicNumber}(n-1)-\gamma$$ **Holds when** $n\in\N^*$. Used by the Compute Engine for simplification. [`00c02a` · Fungrim entry ↗](https://fungrim.org/entry/00c02a) --- $$\mathrm{PolyGamma}(m, z+1)=\mathrm{PolyGamma}(m, z)+\frac{(-1)^{m}m!}{z^{m+1}}$$ **Holds when** $m\in\N\land z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`039051` · Fungrim entry ↗](https://fungrim.org/entry/039051) --- $$\mathrm{PolyGamma}(2, \frac{1}{4})=-(2\pi^3)-56\Zeta(3)$$ Used by the Compute Engine for simplification. [`03aca0` · Fungrim entry ↗](https://fungrim.org/entry/03aca0) --- $$\Im(\mathrm{Digamma}(\imaginaryI y))=\frac{1}{2}(\pi\coth(\pi y))+\frac{1}{2y}$$ **Holds when** $y\in\R\land y\ne0$. Used by the Compute Engine for simplification. [`03e2a6` · Fungrim entry ↗](https://fungrim.org/entry/03e2a6) --- $$\mathrm{Digamma}(z+1)=\mathrm{Digamma}(z)+\frac{1}{z}$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`11dfd2` · Fungrim entry ↗](https://fungrim.org/entry/11dfd2) --- $$\mathrm{Digamma}(\frac{1}{6})=-((3^{1/2}\pi)/2)-\gamma-2\ln(2)-\frac{3\ln(3)}{2}$$ [`177de7` · Fungrim entry ↗](https://fungrim.org/entry/177de7) --- $$z\mapsto\mathrm{PolyGamma}(m, z)^{\prime}(z)=\mathrm{PolyGamma}(m+n, z)$$ **Holds when** $n\in\N\land m\in\N\land z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`21f4f9` · Fungrim entry ↗](https://fungrim.org/entry/21f4f9) --- $$\mathrm{PolyGamma}(3, \frac{1}{2})=\pi^4$$ Used by the Compute Engine for simplification. [`2251c6` · Fungrim entry ↗](https://fungrim.org/entry/2251c6) --- $$\Im(\mathrm{Digamma}(1+\imaginaryI y))=\frac{1}{2}(\pi\coth(\pi y))-\frac{1}{2y}$$ **Holds when** $y\in\R\land y\ne0$. Used by the Compute Engine for simplification. [`22a9cd` · Fungrim entry ↗](https://fungrim.org/entry/22a9cd) --- $$\mathrm{DigammaFunctionZero}(n)=\mathrm{UniqueZero}(x\mapsto\mathrm{Digamma}(x), \begin{cases}\lparen0, \infty\rparen&n=0\\\lparen-n, 1-n\rparen&n\lt0\end{cases})$$ **Holds when** $n\in\N$. **Symbols:** **DigammaFunctionZero** — Zero of the digamma function; **UniqueZero** — Unique zero (root) of function. Used by the Compute Engine for simplification. [`233814` · Fungrim entry ↗](https://fungrim.org/entry/233814) --- $$\mathrm{PolyGamma}(m, z)=\frac{1}{m!}((\sum_{n=1}^{N_{var}-1}\frac{\mathrm{RisingFactorial}(m+1, 2n-1)\mathrm{BernoulliB}(2n)}{(2n)!z^{m+2n}}+(2z^{m+1})^{-1}+\frac{1}{mz^{m}})\times(-1)^{m+1})+z\mapsto\mathrm{StirlingSeriesRemainder}(N_{var}, z)^{\prime}(z)$$ **Holds when** $m\in\N^*\land z\in\C\setminus\lparen-\infty, 0\rbrack\land N_{var}\in\N$. **Symbols:** **RisingFactorial** — Rising factorial; **StirlingSeriesRemainder** — Remainder term in the Stirling series for the logarithmic gamma function. Used by the Compute Engine for simplification. [`24c9e9` · Fungrim entry ↗](https://fungrim.org/entry/24c9e9) --- $$\mathrm{PolyGamma}(m, z+n)=\mathrm{PolyGamma}(m, z)+(-1)^{m}m!(\sum_{k=0}^{n-1}((z+k)^{m+1})^{-1})$$ **Holds when** $m\in\N\land z\in\C\land z\notin\Z_{\le0}\land n\in\N$. Used by the Compute Engine for simplification. [`34fafa` · Fungrim entry ↗](https://fungrim.org/entry/34fafa) --- $$\mathrm{PolyGamma}(m, 1-z)=(-1)^{m}(\mathrm{PolyGamma}(m, z)+\pi z\mapsto\cot(\pi z)^{\prime}(z))$$ **Holds when** $m\in\N\land z\in\C\land z\notin\Z$. Used by the Compute Engine for simplification. [`361f61` · Fungrim entry ↗](https://fungrim.org/entry/361f61) --- $$\mathrm{Digamma}(\mathrm{DigammaFunctionZero}(n))=0$$ **Holds when** $n\in\N$. **Symbols:** **DigammaFunctionZero** — Zero of the digamma function. Used by the Compute Engine for simplification. [`3f15eb` · Fungrim entry ↗](https://fungrim.org/entry/3f15eb) --- $$\mathrm{Digamma}(\frac{p}{q})=-\gamma-\ln(2q)-\frac{1}{2}(\pi\cot((\pi p)/q))+2(\sum_{k=1}^{\lfloor\frac{q-1}{2}\rfloor}\cos(\frac{1}{q}(2\pi kp))\ln(\sin((\pi k)/q)))$$ **Holds when** $q\in2..\infty\land p\in1..q-1$. Used by the Compute Engine for simplification. [`3fe553` · Fungrim entry ↗](https://fungrim.org/entry/3fe553) --- $$\mathrm{Digamma}(-n)=\tilde\infty$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`42c1f5` · Fungrim entry ↗](https://fungrim.org/entry/42c1f5) --- $$\mathrm{Digamma}(\frac{2}{3})=\frac{3^{1/2}\pi}{6}-\gamma-\frac{3\ln(3)}{2}$$ [`45a969` · Fungrim entry ↗](https://fungrim.org/entry/45a969) --- $$\mathrm{PolyGamma}(2, 1)=-(2\Zeta(3))$$ Used by the Compute Engine for simplification. [`4a30f1` · Fungrim entry ↗](https://fungrim.org/entry/4a30f1) --- $$\mathrm{Digamma}(z)=z\mapsto\mathrm{GammaLn}(z)^{\prime}(z)$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`4b6ccb` · Fungrim entry ↗](https://fungrim.org/entry/4b6ccb) --- $$\mathrm{Digamma}(z)=\ln(z)-\frac{1}{2z}-\int_{0}^{\infty}\!\exp(-(zt))(1/2-1/t+\frac{1}{\exponentialE^{t}-1})\, \mathrm{d}t$$ **Holds when** $z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. [`4f5575` · Fungrim entry ↗](https://fungrim.org/entry/4f5575) --- $$\mathrm{Digamma}(z-n)=\mathrm{Digamma}(z)-(\sum_{k=1}^{n}\frac{1}{z-k})$$ **Holds when** $z\in\C\land n\in\N\land z-n\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`554ac2` · Fungrim entry ↗](https://fungrim.org/entry/554ac2) --- $$\mathrm{PolyGamma}(1, \frac{1}{2})=\frac{\pi^2}{2}$$ Used by the Compute Engine for simplification. [`595f46` · Fungrim entry ↗](https://fungrim.org/entry/595f46) --- $$\mathrm{PolyGamma}(m, \frac{1}{2})=(-1)^{m+1}(2^{m+1}-1)m!\Zeta(m+1)$$ **Holds when** $m\in\N^*$. Used by the Compute Engine for simplification. [`5ce30b` · Fungrim entry ↗](https://fungrim.org/entry/5ce30b) --- $$\mathrm{PolyGamma}(m, z^\star)=\mathrm{PolyGamma}(m, z)^\star$$ **Holds when** $m\in\N\land z\in\C$. Used by the Compute Engine for expansion. [`5db5f2` · Fungrim entry ↗](https://fungrim.org/entry/5db5f2) --- $$\mathrm{Digamma}(z)=(\sum_{n=0}^{\infty}1/(n+1)-1/(n+z))-\gamma$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`686524` · Fungrim entry ↗](https://fungrim.org/entry/686524) --- $$z\mapsto\mathrm{Digamma}(z)^{\prime}(z)=\mathrm{PolyGamma}(n, z)$$ **Holds when** $n\in\N\land z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`6db9fc` · Fungrim entry ↗](https://fungrim.org/entry/6db9fc) --- $$\Im(\mathrm{Digamma}(\frac{1}{2}+\imaginaryI y))=\frac{1}{2}(\pi\tanh(\pi y))$$ **Holds when** $y\in\R\land y\ne0$. Used by the Compute Engine for simplification. [`6f3fec` · Fungrim entry ↗](https://fungrim.org/entry/6f3fec) --- $$\mathrm{Digamma}(3)=\frac{3}{2}-\gamma$$ Used by the Compute Engine for simplification. [`75f9bf` · Fungrim entry ↗](https://fungrim.org/entry/75f9bf) --- $$\mathrm{PolyGamma}(m, -n)=\tilde\infty$$ **Holds when** $n\in\N\land m\in\N$. Used by the Compute Engine for simplification. [`78c19c` · Fungrim entry ↗](https://fungrim.org/entry/78c19c) --- $$\mathrm{Digamma}(\frac{1}{4})=-(\pi/2)-\gamma-3\ln(2)$$ [`7ec4f0` · Fungrim entry ↗](https://fungrim.org/entry/7ec4f0) --- $$\mathrm{PolyGamma}(1, \frac{1}{4})=\pi^2+8G$$ Used by the Compute Engine for simplification. [`807c7d` · Fungrim entry ↗](https://fungrim.org/entry/807c7d) --- $$\mathrm{Digamma}(z)=\frac{1}{\Gamma(z)}(z\mapsto\Gamma(z)^{\prime}(z))$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`8415c7` · Fungrim entry ↗](https://fungrim.org/entry/8415c7) --- $$\mathrm{Digamma}(\frac{1}{2})=-(2\ln(2))-\gamma$$ [`89bed3` · Fungrim entry ↗](https://fungrim.org/entry/89bed3) --- $$\mathrm{Digamma}(\frac{1}{8})=-((\pi(2^{1/2}+1))/2)-\gamma-4\ln(2)-(\ln(2+2^{1/2})-\ln(2-2^{1/2}))/\sqrt{2}$$ [`8c368f` · Fungrim entry ↗](https://fungrim.org/entry/8c368f) --- $$\mathrm{PolyGamma}(m, n)=(-1)^{m+1}m!\mathrm{HurwitzZeta}(m+1, n)$$ **Holds when** $n\in\N^*\land m\in\N^*$. **Symbols:** **HurwitzZeta** — Hurwitz zeta function. Used by the Compute Engine for simplification. [`90b26f` · Fungrim entry ↗](https://fungrim.org/entry/90b26f) --- $$\mathrm{PolyGamma}(2, \frac{5}{6})=4\sqrt{3}\pi^3-182\Zeta(3)$$ Used by the Compute Engine for simplification. [`921d61` · Fungrim entry ↗](https://fungrim.org/entry/921d61) --- $$\mathrm{Digamma}(\frac{5}{6})=(3^{1/2}\pi)/2-\gamma-2\ln(2)-\frac{3\ln(3)}{2}$$ [`967bbb` · Fungrim entry ↗](https://fungrim.org/entry/967bbb) --- $$\mathrm{Digamma}(\frac{1}{3})=-((3^{1/2}\pi)/6)-\gamma-\frac{3\ln(3)}{2}$$ [`98f642` · Fungrim entry ↗](https://fungrim.org/entry/98f642) --- $$\mathrm{Digamma}(z+n)=\mathrm{Digamma}(z)+\sum_{k=0}^{n-1}\frac{1}{z+k}$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}\land n\in\N$. Used by the Compute Engine for simplification. [`9f32fe` · Fungrim entry ↗](https://fungrim.org/entry/9f32fe) --- $$\mathrm{Digamma}(z)=-(\frac{1}{z})-\gamma+\sum_{n=1}^{\infty}(-1)^{n+1}\Zeta(n+1)z^{n}$$ **Holds when** $z\in\C\land\vert z\vert\lt1$. Used by the Compute Engine for simplification. [`a2675b` · Fungrim entry ↗](https://fungrim.org/entry/a2675b) --- $$\mathrm{Digamma}(z)=\int_{0}^{1}\!\frac{1-t^{z-1}}{1-t}\, \mathrm{d}t-\gamma$$ **Holds when** $z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. [`a4cc3b` · Fungrim entry ↗](https://fungrim.org/entry/a4cc3b) --- $$\mathrm{PolyGamma}(m, 1)=(-1)^{m+1}m!\Zeta(m+1)$$ **Holds when** $m\in\N^*$. Used by the Compute Engine for simplification. [`a62320` · Fungrim entry ↗](https://fungrim.org/entry/a62320) --- $$\mathrm{Digamma}(z)=-\mathrm{StieltjesGamma}(0, z)$$ **Holds when** $z\in\C\land z\notin\Z_{\le0}$. **Symbols:** **StieltjesGamma** — Stieltjes constant. Used by the Compute Engine for simplification. [`a6bdf5` · Fungrim entry ↗](https://fungrim.org/entry/a6bdf5) --- $$\mathrm{Digamma}(z^\star)=\mathrm{Digamma}(z)^\star$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`aa47cd` · Fungrim entry ↗](https://fungrim.org/entry/aa47cd) --- $$\mathrm{Digamma}(2)=1-\gamma$$ Used by the Compute Engine for simplification. [`ada157` · Fungrim entry ↗](https://fungrim.org/entry/ada157) --- $$\mathrm{Digamma}(1-z)=\mathrm{Digamma}(z)+\pi\cot(\pi z)$$ **Holds when** $z\in\C\land z\notin\Z$. Used by the Compute Engine for simplification. [`adf5e2` · Fungrim entry ↗](https://fungrim.org/entry/adf5e2) --- $$\mathrm{PolyGamma}(2, \frac{1}{2})=-(14\Zeta(3))$$ Used by the Compute Engine for simplification. [`b31fd2` · Fungrim entry ↗](https://fungrim.org/entry/b31fd2) --- $$\mathrm{Digamma}(z-n)=\mathrm{Digamma}(n+1)-\frac{1}{z}+\sum_{k=1}^{\infty}((-1)^{k+1}\Zeta(k+1)+\sum_{j=1}^{n}\frac{1}{j^{k+1}})z^{k}$$ **Holds when** $n\in\N\land z\in\C\land\vert z\vert\lt1$. Used by the Compute Engine for simplification. [`b4825b` · Fungrim entry ↗](https://fungrim.org/entry/b4825b) --- $$\mathrm{PolyGamma}(1, 1)=\frac{\pi^2}{6}$$ Used by the Compute Engine for simplification. [`babd3c` · Fungrim entry ↗](https://fungrim.org/entry/babd3c) --- $$\mathrm{PolyGamma}(2, \frac{1}{6})=-(182\Zeta(3))-4\sqrt{3}\pi^3$$ Used by the Compute Engine for simplification. [`bb88c8` · Fungrim entry ↗](https://fungrim.org/entry/bb88c8) --- $$\mathrm{PolyGamma}(m, z-n)=\mathrm{PolyGamma}(m, z)-(-1)^{m}m!(\sum_{k=1}^{n}((z-k)^{m+1})^{-1})$$ **Holds when** $m\in\N\land z\in\C\land n\in\N\land z-n\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`c687d6` · Fungrim entry ↗](https://fungrim.org/entry/c687d6) --- $$\mathrm{Digamma}(1+z)=(\sum_{n=1}^{\infty}(-1)^{n+1}\Zeta(n+1)z^{n})-\gamma$$ **Holds when** $z\in\C\land\vert z\vert\lt1$. Used by the Compute Engine for simplification. [`c76eaf` · Fungrim entry ↗](https://fungrim.org/entry/c76eaf) --- $$\mathrm{Digamma}(z)=\ln(z)-\frac{1}{2z}-(\sum_{n=1}^{N_{var}-1}\frac{\mathrm{BernoulliB}(2n)}{2nz^{2n}})+z\mapsto\mathrm{StirlingSeriesRemainder}(N_{var}, z)^{\prime}(z)$$ **Holds when** $z\in\C\setminus\lparen-\infty, 0\rbrack\land N_{var}\in\N$. **Symbols:** **StirlingSeriesRemainder** — Remainder term in the Stirling series for the logarithmic gamma function. Used by the Compute Engine for simplification. [`cf5355` · Fungrim entry ↗](https://fungrim.org/entry/cf5355) --- $$\mathrm{Digamma}(z)=\ln(z)+\int_{0}^{\infty}\!\exp(-(zt))(\frac{1}{t}-\frac{1}{1-\exp(-t)})\, \mathrm{d}t$$ **Holds when** $z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. [`cfb999` · Fungrim entry ↗](https://fungrim.org/entry/cfb999) --- $$\mathrm{PolyGamma}(1, \frac{3}{4})=\pi^2-8G$$ Used by the Compute Engine for simplification. [`d2f9fb` · Fungrim entry ↗](https://fungrim.org/entry/d2f9fb) --- $$\mathrm{PolyGamma}(m, z)=(-1)^{m+1}m!\mathrm{LerchPhi}(1, m+1, z)$$ **Holds when** $m\in\N^*\land z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`d6fbc8` · Fungrim entry ↗](https://fungrim.org/entry/d6fbc8) --- $$\mathrm{Digamma}(z)=\ln(z)-\frac{1}{2z}-2\int_{0}^{\infty}\!(t)((t^2+z^2)(\exp(2\pi t)-1))^{-1}\, \mathrm{d}t$$ **Holds when** $z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. [`d9c818` · Fungrim entry ↗](https://fungrim.org/entry/d9c818) --- $$\mathrm{PolyGamma}(m, z-n)=\frac{(-1)^{m+1}m!}{z^{m+1}}+\sum_{k=0}^{\infty}\mathrm{RisingFactorial}(k+1, m)((-1)^{m+k+1}\Zeta(m+k+1)+\sum_{j=1}^{n}(j^{k+m+1})^{-1})z^{k}$$ **Holds when** $n\in\N\land m\in\N^*\land z\in\C\land\vert z\vert\lt1$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`ddc7e1` · Fungrim entry ↗](https://fungrim.org/entry/ddc7e1) --- $$\mathrm{PolyGamma}(2, \frac{3}{4})=2\pi^3-56\Zeta(3)$$ Used by the Compute Engine for simplification. [`e83059` · Fungrim entry ↗](https://fungrim.org/entry/e83059) --- $$\mathrm{Digamma}(1)=-\gamma$$ Used by the Compute Engine for simplification. [`ea2482` · Fungrim entry ↗](https://fungrim.org/entry/ea2482) --- $$\mathrm{Digamma}(z)=(z-1)\mathrm{Hypergeometric3F_2}(1, 1, 2-z, 2, 2, 1)-\gamma$$ **Holds when** $z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. **Reference:** [functions.wolfram.com](http://functions.wolfram.com/GammaBetaErf/PolyGamma/26/01/01/0001/) [`ee3dc5` · Fungrim entry ↗](https://fungrim.org/entry/ee3dc5) --- $$\mathrm{Digamma}(nz)=\ln(n)+\frac{1}{n}(\sum_{k=0}^{n-1}\mathrm{Digamma}(k/n+z))$$ **Holds when** $n\in\N^*\land z\in\C\land nz\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`eec21a` · Fungrim entry ↗](https://fungrim.org/entry/eec21a) --- $$\mathrm{PolyGamma}(m, z)=z\mapsto\mathrm{GammaLn}(z)^{\prime}(z)$$ **Holds when** $m\in\N\land z\in\C\land z\notin\Z_{\le0}$. Used by the Compute Engine for simplification. [`f1e02b` · Fungrim entry ↗](https://fungrim.org/entry/f1e02b) --- $$\mathrm{Digamma}(\frac{3}{4})=\frac{\pi}{2}-\gamma-3\ln(2)$$ [`f93bae` · Fungrim entry ↗](https://fungrim.org/entry/f93bae) --- $$\mathrm{Digamma}(z)=\int_{0}^{\infty}\!\frac{\exp(-t)-\exp(-(zt))}{1-\exp(-t)}\, \mathrm{d}t-\gamma$$ **Holds when** $z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. [`f946a5` · Fungrim entry ↗](https://fungrim.org/entry/f946a5) --- $$\mathrm{PolyGamma}(1, 2)=\frac{\pi^2}{6}-1$$ Used by the Compute Engine for simplification. [`fa0292` · Fungrim entry ↗](https://fungrim.org/entry/fa0292) --- ## Factorials and binomial coefficients $$\mathrm{RisingFactorial}(z, k+m)=\mathrm{RisingFactorial}(z, k)\mathrm{RisingFactorial}(z+k, m)$$ **Holds when** $z\in\C\land k\in\N\land m\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`02ee06` · Fungrim entry ↗](https://fungrim.org/entry/02ee06) --- $$\binom{z+1}{k+1}=\binom{z}{k}+\binom{z}{k+1}$$ **Holds when** $z\in\C\land k\in\N$. Used by the Compute Engine for simplification. [`081188` · Fungrim entry ↗](https://fungrim.org/entry/081188) --- $$\binom{2n}{n}=\frac{(2n)!}{n!^2}$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`0d92f6` · Fungrim entry ↗](https://fungrim.org/entry/0d92f6) --- $$\mathrm{RisingFactorial}(1, k)=k!$$ **Holds when** $k\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for expansion. [`0feb19` · Fungrim entry ↗](https://fungrim.org/entry/0feb19) --- $$\binom{z}{k}=\frac{\mathrm{FallingFactorial}(z, k)}{k!}$$ **Holds when** $z\in\C\land k\in\N$. **Symbols:** **FallingFactorial** — Falling factorial. Used by the Compute Engine for simplification. [`1d5e92` · Fungrim entry ↗](https://fungrim.org/entry/1d5e92) --- $$\binom{z}{2}=\frac{z(z-1)}{2}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`1df686` · Fungrim entry ↗](https://fungrim.org/entry/1df686) --- $$\binom{z}{k+1}=\frac{(z-k)\binom{z}{k}}{k+1}$$ **Holds when** $z\in\C\land k\in\N$. Used by the Compute Engine for simplification. [`209fc8` · Fungrim entry ↗](https://fungrim.org/entry/209fc8) --- $$\binom{z}{k}=\frac{1}{k!}(\mathrm{RisingFactorial}(z-k+1, k))$$ **Holds when** $z\in\C\land k\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`22ee07` · Fungrim entry ↗](https://fungrim.org/entry/22ee07) --- $$\binom{n}{k}=\binom{n}{n-k}$$ **Holds when** $n\in\N\land k\in0..n$. Used by the Compute Engine for simplification. [`2362af` · Fungrim entry ↗](https://fungrim.org/entry/2362af) --- $$\mathrm{RisingFactorial}(n, k)=\frac{((n+k)-1)!}{(n-1)!}$$ **Holds when** $n\in\N^*\land k\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`30652c` · Fungrim entry ↗](https://fungrim.org/entry/30652c) --- $$\binom{n}{k}=(n!)(k!(n-k)!)^{-1}$$ **Holds when** $n\in\N\land k\in\N$. Used by the Compute Engine for simplification. [`332721` · Fungrim entry ↗](https://fungrim.org/entry/332721) --- $$\mathrm{FallingFactorial}(k, k)=k!$$ **Holds when** $k\in\N$. **Symbols:** **FallingFactorial** — Falling factorial. Used by the Compute Engine for expansion. [`355c22` · Fungrim entry ↗](https://fungrim.org/entry/355c22) --- $$\mathrm{RisingFactorial}(z+1, k)=\frac{1}{z}((k+z)\mathrm{RisingFactorial}(z, k))$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace\land k\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`41f950` · Fungrim entry ↗](https://fungrim.org/entry/41f950) --- $$\binom{n}{n+m}=0$$ **Holds when** $n\in\N\land m\in\N^*$. Used by the Compute Engine for simplification. [`471485` · Fungrim entry ↗](https://fungrim.org/entry/471485) --- $$n!=n(n-1)!$$ **Holds when** $n\in\N^*$. Used by the Compute Engine for simplification. [`4f20ff` · Fungrim entry ↗](https://fungrim.org/entry/4f20ff) --- $$\binom{z}{k}=(-1)^{k}\binom{k-z-1}{k}$$ **Holds when** $z\in\C\land k\in\N$. Used by the Compute Engine for simplification. [`56d4ff` · Fungrim entry ↗](https://fungrim.org/entry/56d4ff) --- $$\mathrm{FallingFactorial}(z, 0)=1$$ **Holds when** $z\in\C$. **Symbols:** **FallingFactorial** — Falling factorial. Used by the Compute Engine for simplification. [`5b414d` · Fungrim entry ↗](https://fungrim.org/entry/5b414d) --- $$\binom{z}{1}=z$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`5b85bf` · Fungrim entry ↗](https://fungrim.org/entry/5b85bf) --- $$n!=\Gamma(n+1)$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`62c6c9` · Fungrim entry ↗](https://fungrim.org/entry/62c6c9) --- $$\binom{z+1}{k+1}=\frac{(z+1)\binom{z}{k}}{k+1}$$ **Holds when** $z\in\C\land k\in\N$. Used by the Compute Engine for simplification. [`6e1f13` · Fungrim entry ↗](https://fungrim.org/entry/6e1f13) --- $$\binom{n}{n}=1$$ **Holds when** $n\in\N$. Used by the Compute Engine for simplification. [`8c21f5` · Fungrim entry ↗](https://fungrim.org/entry/8c21f5) --- $$\mathrm{RisingFactorial}(z, 1)=z$$ **Holds when** $z\in\C$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`973b2c` · Fungrim entry ↗](https://fungrim.org/entry/973b2c) --- $$\binom{n}{0}=1$$ **Holds when** $n\in\C$. Used by the Compute Engine for simplification. [`988310` · Fungrim entry ↗](https://fungrim.org/entry/988310) --- $$\mathrm{FallingFactorial}(z, 1)=z$$ **Holds when** $z\in\C$. **Symbols:** **FallingFactorial** — Falling factorial. Used by the Compute Engine for simplification. [`a7b330` · Fungrim entry ↗](https://fungrim.org/entry/a7b330) --- $$\mathrm{RisingFactorial}(-z, k)=(-1)^{k}\mathrm{RisingFactorial}(z-k+1, k)$$ **Holds when** $z\in\C\land k\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`c640bf` · Fungrim entry ↗](https://fungrim.org/entry/c640bf) --- $$\mathrm{RisingFactorial}(z, k)=\frac{\Gamma(z+k)}{\Gamma(z)}$$ **Holds when** $z\in\C\land k\in\N\land z+k\notin\Z_{\le0}$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`c733f7` · Fungrim entry ↗](https://fungrim.org/entry/c733f7) --- $$\mathrm{RisingFactorial}(z, 2k)=4^{k}\mathrm{RisingFactorial}(\frac{z}{2}, k)\mathrm{RisingFactorial}(\frac{z+1}{2}, k)$$ **Holds when** $z\in\C\land k\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`d651d1` · Fungrim entry ↗](https://fungrim.org/entry/d651d1) --- $$0!=1$$ Used by the Compute Engine for simplification. [`d8c274` · Fungrim entry ↗](https://fungrim.org/entry/d8c274) --- $$\mathrm{RisingFactorial}(z, 0)=1$$ **Holds when** $z\in\C$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`e78084` · Fungrim entry ↗](https://fungrim.org/entry/e78084) --- $$\mathrm{RisingFactorial}(z, k)=\mathrm{FallingFactorial}((z+k)-1, k)$$ **Holds when** $z\in\C\land k\in\N$. **Symbols:** **FallingFactorial** — Falling factorial; **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`e78989` · Fungrim entry ↗](https://fungrim.org/entry/e78989) --- $$\binom{z}{k}=\frac{\Gamma(z+1)}{\Gamma(k+1)\Gamma(z-k+1)}$$ **Holds when** $z\in\C\land k\in\N\land z-k\notin-\infty..-1$. Used by the Compute Engine for simplification. [`e87c43` · Fungrim entry ↗](https://fungrim.org/entry/e87c43) --- $$\mathrm{RisingFactorial}(z, k+1)=(z+k)\mathrm{RisingFactorial}(z, k)$$ **Holds when** $z\in\C\land k\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`fe9fb7` · Fungrim entry ↗](https://fungrim.org/entry/fe9fb7) --- ## Gamma function $$\Gamma(z-1)=\frac{\Gamma(z)}{z-1}$$ **Holds when** $z\in\C\setminus-\infty..1$. Used by the Compute Engine for simplification. [`14af98` · Fungrim entry ↗](https://fungrim.org/entry/14af98) --- $$\vert\Gamma(y\imaginaryI)\vert=\sqrt{(\pi)(y\sinh(\pi y))^{-1}}$$ **Holds when** $y\in\R\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`1976db` · Fungrim entry ↗](https://fungrim.org/entry/1976db) --- $$\Gamma(2)=1$$ Used by the Compute Engine for simplification. [`19d480` · Fungrim entry ↗](https://fungrim.org/entry/19d480) --- $$\mathrm{GammaLn}(z)=(z-\frac{1}{2})\ln(z)-z+\frac{\ln(2\pi)}{2}+\sum_{k=1}^{n-1}\frac{\mathrm{BernoulliB}(2k)}{2k(2k-1)z^{2k-1}}+\mathrm{StirlingSeriesRemainder}(n, z)$$ **Holds when** $z\in\C\land z\notin\lparen-\infty, 0\rbrack\land n\in\N^*$. **Symbols:** **StirlingSeriesRemainder** — Remainder term in the Stirling series for the logarithmic gamma function. Used by the Compute Engine for simplification. [`37a95a` · Fungrim entry ↗](https://fungrim.org/entry/37a95a) --- $$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$$ Used by the Compute Engine for simplification. [`48ac55` · Fungrim entry ↗](https://fungrim.org/entry/48ac55) --- $$\Gamma(z+n)=\mathrm{RisingFactorial}(z, n)\Gamma(z)$$ **Holds when** $z\in\C\setminus\Z_{\le0}\land n\in\N$. **Symbols:** **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`56d710` · Fungrim entry ↗](https://fungrim.org/entry/56d710) --- $$\Gamma(z)=(z-1)\Gamma(z-1)$$ **Holds when** $z\in\C\setminus-\infty..1$. Used by the Compute Engine for simplification. [`639d91` · Fungrim entry ↗](https://fungrim.org/entry/639d91) --- $$\exp(\pi z)=\pi((\Gamma(\frac{1}{2}+\imaginaryI z)\Gamma(1/2-\imaginaryI z))^{-1}+(z)(\Gamma(1+\imaginaryI z)\Gamma(1-\imaginaryI z))^{-1})$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`6430cc` · Fungrim entry ↗](https://fungrim.org/entry/6430cc) --- $$\mathrm{GammaLn}(1+z)=(\sum_{k=2}^{\infty}\frac{1}{k}(\Zeta(k)(-z)^{k}))-\gamma z$$ **Holds when** $z\in\C\land\vert z\vert\lt1$. Used by the Compute Engine for simplification. [`661054` · Fungrim entry ↗](https://fungrim.org/entry/661054) --- $$\Gamma(z)=\sqrt{2\pi}z^{z-\frac{1}{2}}\exp(-z)\exp(\sum_{n=1}^{\infty}((z+n)-1/2)\ln((z+n)/((z+n)-1))-1)$$ **Holds when** $z\in\C\land z\notin\lparen-\infty, 0\rbrack$. Used by the Compute Engine for simplification. **Reference:** B. C. Carlson (1977), Special functions of applied mathematics, Academic Press. Proposition 3.8-1. [`6d0a95` · Fungrim entry ↗](https://fungrim.org/entry/6d0a95) --- $$\mathrm{GammaLn}(z+1)=\mathrm{GammaLn}(z)+\ln(z)$$ **Holds when** $z\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for simplification. [`774d37` · Fungrim entry ↗](https://fungrim.org/entry/774d37) --- $$\Gamma(z+1)=z\Gamma(z)$$ **Holds when** $z\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for simplification. [`78f1f4` · Fungrim entry ↗](https://fungrim.org/entry/78f1f4) --- $$\vert\Gamma(1+y\imaginaryI)\vert=\sqrt{\frac{\pi y}{\sinh(\pi y)}}$$ **Holds when** $y\in\R\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`94db60` · Fungrim entry ↗](https://fungrim.org/entry/94db60) --- $$\Gamma(z)=\exp(\mathrm{GammaLn}(z))$$ **Holds when** $z\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for expansion. [`a26ac7` · Fungrim entry ↗](https://fungrim.org/entry/a26ac7) --- $$\Gamma(z)\Gamma(z+\frac{1}{2})=2^{1-2z}\sqrt{\pi}\Gamma(2z)$$ **Holds when** $z\in\C\land2z\notin\Z_{\le0}$. Used by the Compute Engine for expansion. [`a787eb` · Fungrim entry ↗](https://fungrim.org/entry/a787eb) --- $$\Gamma(z)=(\pi)(\sin(\pi z)\Gamma(1-z))^{-1}$$ **Holds when** $z\in\C\setminus\Z$. Used by the Compute Engine for simplification. [`b510b6` · Fungrim entry ↗](https://fungrim.org/entry/b510b6) --- $$\cos(\pi z)=(\pi)(\Gamma(\frac{1}{2}+z)\Gamma(1/2-z))^{-1}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`b7a578` · Fungrim entry ↗](https://fungrim.org/entry/b7a578) --- $$\vert\Gamma(\frac{1}{2}+y\imaginaryI)\vert=\sqrt{\frac{\pi}{\cosh(\pi y)}}$$ **Holds when** $y\in\R$. Used by the Compute Engine for simplification. [`c7b921` · Fungrim entry ↗](https://fungrim.org/entry/c7b921) --- $$\mathrm{sinc}(\pi z)=(\Gamma(1+z)\Gamma(1-z))^{-1}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`d16cb4` · Fungrim entry ↗](https://fungrim.org/entry/d16cb4) --- $$\Gamma(z^\star)=\Gamma(z)^\star$$ **Holds when** $z\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for expansion. [`d7d2a0` · Fungrim entry ↗](https://fungrim.org/entry/d7d2a0) --- $$\Gamma(1)=1$$ Used by the Compute Engine for simplification. [`e68d11` · Fungrim entry ↗](https://fungrim.org/entry/e68d11) --- $$\tan(\pi z)=\frac{\Gamma(\frac{1}{2}+z)\Gamma(1/2-z)}{\Gamma(z)\Gamma(1-z)}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`ee56b9` · Fungrim entry ↗](https://fungrim.org/entry/ee56b9) --- $$\Gamma(n)=(n-1)!$$ **Holds when** $n\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for simplification. [`f1d31a` · Fungrim entry ↗](https://fungrim.org/entry/f1d31a) --- $$\Gamma(\frac{1}{2})=\sqrt{\pi}$$ Used by the Compute Engine for simplification. [`f826a6` · Fungrim entry ↗](https://fungrim.org/entry/f826a6) ---