Gamma and related functions

Part of the Fungrim Identities reference — 135 identities for gamma and related functions.

Generated reference

This page is generated from the compiled Fungrim artifact by scripts/fungrim/gen-reference-doc.ts (upstream snapshot 953c2afd2822, translator grim2mathjson 0.1.0). Do not edit it by hand. The corpus is MIT-licensed; see data/fungrim/LICENSE.

Contents

• Barnes G-function (27)
• Beta function (13)
• Digamma function (39)
• Factorials and binomial coefficients (32)
• 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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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(\mathrm{PolyLog}(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 541e2e · Fungrim entry ↗

---

\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 ↗

---

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 ↗

---

\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 6395ee · Fungrim entry ↗

---

\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 ↗

---

\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: BernoulliB — Bernoulli number; 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 6f8e14 · Fungrim entry ↗

---

\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)\mathrm{PolyLog}(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)\mathrm{PolyLog}(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 ↗

---

\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 ↗

---

\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 ↗

---

\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 8c96a5 · Fungrim entry ↗

---

\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 95f771 · Fungrim entry ↗

---

\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 ↗

---

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 ↗

---

\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 ↗

---

\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 ↗

---

\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 b64782 · Fungrim entry ↗

---

\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 ↗

---

\mathrm{LogBarnesG}(1-x)=\mathrm{LogBarnesG}(1+x)+x\ln(\frac{\vert\sin(\pi x)\vert}{\pi})+\frac{\Im(\mathrm{PolyLog}(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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

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 ↗

---

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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

(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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

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 ↗

---

\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 ↗

---

\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 ↗

---

\mathrm{Digamma}(\frac{1}{6})=-((3^{1/2}\pi)/2)-\gamma-2\ln(2)-\frac{3\ln(3)}{2}

177de7 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\mathrm{Digamma}(-n)=\tilde\infty

Holds when n\in\N. Used by the Compute Engine for simplification. 42c1f5 · Fungrim entry ↗

---

\mathrm{Digamma}(\frac{2}{3})=\frac{3^{1/2}\pi}{6}-\gamma-\frac{3\ln(3)}{2}

45a969 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\mathrm{Digamma}(3)=\frac{3}{2}-\gamma

Used by the Compute Engine for simplification. 75f9bf · Fungrim entry ↗

---

\mathrm{Digamma}(\frac{1}{4})=-(\pi/2)-\gamma-3\ln(2)

7ec4f0 · Fungrim entry ↗

---

\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 ↗

---

\mathrm{Digamma}(\frac{1}{2})=-(2\ln(2))-\gamma

89bed3 · Fungrim entry ↗

---

\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 ↗

---

\mathrm{Digamma}(\frac{5}{6})=(3^{1/2}\pi)/2-\gamma-2\ln(2)-\frac{3\ln(3)}{2}

967bbb · Fungrim entry ↗

---

\mathrm{Digamma}(\frac{1}{3})=-((3^{1/2}\pi)/6)-\gamma-\frac{3\ln(3)}{2}

98f642 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\mathrm{Digamma}(z^\star)=\mathrm{Digamma}(z)^\star

Holds when z\in\C. Used by the Compute Engine for expansion. aa47cd · Fungrim entry ↗

---

\mathrm{Digamma}(2)=1-\gamma

Used by the Compute Engine for simplification. ada157 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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: BernoulliB — Bernoulli number; StirlingSeriesRemainder — Remainder term in the Stirling series for the logarithmic gamma function. Used by the Compute Engine for simplification. cf5355 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\mathrm{Digamma}(1)=-\gamma

Used by the Compute Engine for simplification. ea2482 · Fungrim entry ↗

---

\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 ee3dc5 · Fungrim entry ↗

---

\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 ↗

---

\mathrm{Digamma}(\frac{3}{4})=\frac{\pi}{2}-\gamma-3\ln(2)

f93bae · Fungrim entry ↗

---

\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 ↗

---

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 ↗

---

\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 ↗

---

\binom{2n}{n}=\frac{(2n)!}{n!^2}

Holds when n\in\N. Used by the Compute Engine for simplification. 0d92f6 · Fungrim entry ↗

---

\mathrm{RisingFactorial}(1, k)=k!

Holds when k\in\N. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for expansion. 0feb19 · Fungrim entry ↗

---

\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 ↗

---

\binom{z}{2}=\frac{z(z-1)}{2}

Holds when z\in\C. Used by the Compute Engine for simplification. 1df686 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\mathrm{FallingFactorial}(k, k)=k!

Holds when k\in\N. Symbols: FallingFactorial — Falling factorial. Used by the Compute Engine for expansion. 355c22 · Fungrim entry ↗

---

\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 ↗

---

\binom{n}{n+m}=0

Holds when n\in\N\land m\in\N^*. Used by the Compute Engine for simplification. 471485 · Fungrim entry ↗

---

n!=n(n-1)!

Holds when n\in\N^*. Used by the Compute Engine for simplification. 4f20ff · Fungrim entry ↗

---

\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 ↗

---

\mathrm{FallingFactorial}(z, 0)=1

Holds when z\in\C. Symbols: FallingFactorial — Falling factorial. Used by the Compute Engine for simplification. 5b414d · Fungrim entry ↗

---

\binom{z}{1}=z

Holds when z\in\C. Used by the Compute Engine for simplification. 5b85bf · Fungrim entry ↗

---

n!=\Gamma(n+1)

Holds when n\in\N. Used by the Compute Engine for simplification. 62c6c9 · Fungrim entry ↗

---

\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 ↗

---

\binom{n}{n}=1

Holds when n\in\N. Used by the Compute Engine for simplification. 8c21f5 · Fungrim entry ↗

---

\mathrm{RisingFactorial}(z, 1)=z

Holds when z\in\C. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. 973b2c · Fungrim entry ↗

---

\binom{n}{0}=1

Holds when n\in\C. Used by the Compute Engine for simplification. 988310 · Fungrim entry ↗

---

\mathrm{FallingFactorial}(z, 1)=z

Holds when z\in\C. Symbols: FallingFactorial — Falling factorial. Used by the Compute Engine for simplification. a7b330 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

0!=1

Used by the Compute Engine for simplification. d8c274 · Fungrim entry ↗

---

\mathrm{RisingFactorial}(z, 0)=1

Holds when z\in\C. Symbols: RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. e78084 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

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 ↗

---

\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 ↗

---

\Gamma(2)=1

Used by the Compute Engine for simplification. 19d480 · Fungrim entry ↗

---

\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: BernoulliB — Bernoulli number; StirlingSeriesRemainder — Remainder term in the Stirling series for the logarithmic gamma function. Used by the Compute Engine for simplification. 37a95a · Fungrim entry ↗

---

\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}

Used by the Compute Engine for simplification. 48ac55 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\Gamma(z+1)=z\Gamma(z)

Holds when z\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. 78f1f4 · Fungrim entry ↗

---

\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 ↗

---

\Gamma(z)=\exp(\mathrm{GammaLn}(z))

Holds when z\in\C\setminus\Z_{\le0}. Used by the Compute Engine for expansion. a26ac7 · Fungrim entry ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\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 ↗

---

\Gamma(z^\star)=\Gamma(z)^\star

Holds when z\in\C\setminus\Z_{\le0}. Used by the Compute Engine for expansion. d7d2a0 · Fungrim entry ↗

---

\Gamma(1)=1

Used by the Compute Engine for simplification. e68d11 · Fungrim entry ↗

---

\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 ↗

---

\Gamma(n)=(n-1)!

Holds when n\in\C\setminus\Z_{\le0}. Used by the Compute Engine for simplification. f1d31a · Fungrim entry ↗

---

\Gamma(\frac{1}{2})=\sqrt{\pi}

Used by the Compute Engine for simplification. f826a6 · Fungrim entry ↗

---