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Digamma function

Table of contents: Definitions - Illustrations - Domain and singularities - Specific values - Zeros - Derivatives and differential equations - Series representations - Representation by other functions - Functional equations - Integral representations - Generating functions - Finite sums - Bounds and inequalities - Summation of rational functions

Definitions

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Symbol: DigammaFunction — ψ ⁣(z)\psi\!\left(z\right) — Digamma function
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Symbol: DigammaFunctionZero — xnx_{n} — Zero of the digamma function

Illustrations

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Image: X-ray of ψ ⁣(z)\psi\!\left(z\right) on z∈[−5,5]+[−5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i
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Image: X-ray of ψ′ ⁣(z)\psi'\!\left(z\right) on z∈[−5,5]+[−5,5]iz \in \left[-5, 5\right] + \left[-5, 5\right] i

Domain and singularities

Digamma function

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x∈R∖{0,−1,…}    ⟹    ψ ⁣(x)∈Rx \in \mathbb{R} \setminus \{0, -1, \ldots\} \;\implies\; \psi\!\left(x\right) \in \mathbb{R}
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z∈C∖{0,−1,…}    ⟹    ψ ⁣(z)∈Cz \in \mathbb{C} \setminus \{0, -1, \ldots\} \;\implies\; \psi\!\left(z\right) \in \mathbb{C}
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ψ ⁣(z) is holomorphic on z∈C∖{0,−1,…}\psi\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \{0, -1, \ldots\}
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ψ ⁣(z) is meromorphic on z∈C\psi\!\left(z\right) \text{ is meromorphic on } z \in \mathbb{C}
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poles⁡ z∈Cψ ⁣(z)={0,−1,…}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \psi\!\left(z\right) = \{0, -1, \ldots\}

Polygamma functions

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ψ(0) ⁣(z)=ψ ⁣(z)\psi^{(0)}\!\left(z\right) = \psi\!\left(z\right)
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(m∈Z≥0  and⁡  x∈R∖{0,−1,…})    ⟹    ψ(m) ⁣(x)∈R\left(m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \;\implies\; \psi^{(m)}\!\left(x\right) \in \mathbb{R}
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(m∈Z≥0  and⁡  z∈C∖{0,−1,…})    ⟹    ψ(m) ⁣(z)∈C\left(m \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \;\implies\; \psi^{(m)}\!\left(z\right) \in \mathbb{C}
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ψ(m) ⁣(z) is holomorphic on z∈C∖{0,−1,…}\psi^{(m)}\!\left(z\right) \text{ is holomorphic on } z \in \mathbb{C} \setminus \{0, -1, \ldots\}
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ψ(m) ⁣(z) is meromorphic on z∈C\psi^{(m)}\!\left(z\right) \text{ is meromorphic on } z \in \mathbb{C}
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poles⁡ z∈Cψ(m) ⁣(z)={0,−1,…}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C}} \psi^{(m)}\!\left(z\right) = \{0, -1, \ldots\}

Specific values

Main topic: Specific values of the digamma function

Zeros

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Table of xnx_{n} to 50 digits for 0≤n≤100 \le n \le 10
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ψ ⁣(xn)=0\psi\!\left(x_{n}\right) = 0

Values at integers and simple fractions

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ψ ⁣(1)=−γ\psi\!\left(1\right) = -\gamma
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ψ ⁣(2)=1−γ\psi\!\left(2\right) = 1 - \gamma
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ψ ⁣(n)=Hn−1−γ\psi\!\left(n\right) = H_{n - 1} - \gamma
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ψ ⁣(12)=−2log⁡(2)−γ\psi\!\left(\frac{1}{2}\right) = -2 \log(2) - \gamma
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ψ ⁣(13)=−3π6−γ−3log⁡(3)2\psi\!\left(\frac{1}{3}\right) = -\frac{\sqrt{3} \pi}{6} - \gamma - \frac{3 \log(3)}{2}
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ψ ⁣(14)=−π2−γ−3log⁡(2)\psi\!\left(\frac{1}{4}\right) = -\frac{\pi}{2} - \gamma - 3 \log(2)
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ψ ⁣(pq)=−γ−log⁡ ⁣(2q)−π2cot⁡ ⁣(πpq)+2∑k=1⌊(q−1)/2⌋cos⁡ ⁣(2πkpq)log⁡ ⁣(sin⁡ ⁣(πkq))\psi\!\left(\frac{p}{q}\right) = -\gamma - \log\!\left(2 q\right) - \frac{\pi}{2} \cot\!\left(\frac{\pi p}{q}\right) + 2 \sum_{k=1}^{\left\lfloor \left( q - 1 \right) / 2 \right\rfloor} \cos\!\left(\frac{2 \pi k p}{q}\right) \log\!\left(\sin\!\left(\frac{\pi k}{q}\right)\right)

Values of polygamma functions at integers and simple fractions

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ψ′ ⁣(1)=π26\psi'\!\left(1\right) = \frac{{\pi}^{2}}{6}
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ψ(m) ⁣(n)=(−1)m+1m!ζ ⁣(m+1,n)\psi^{(m)}\!\left(n\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, n\right)
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ψ(m) ⁣(12)=(−1)m+1(2m+1−1)m!ζ ⁣(m+1)\psi^{(m)}\!\left(\frac{1}{2}\right) = {\left(-1\right)}^{m + 1} \left({2}^{m + 1} - 1\right) m ! \zeta\!\left(m + 1\right)
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ψ′ ⁣(14)=π2+8G\psi'\!\left(\frac{1}{4}\right) = {\pi}^{2} + 8 G

Zeros

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n∈Z≥0    ⟹    xn∈Rn \in \mathbb{Z}_{\ge 0} \;\implies\; x_{n} \in \mathbb{R}
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zeros⁡ z∈Cψ ⁣(z)={xn:n∈Z≥0}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \psi\!\left(z\right) = \left\{ x_{n} : n \in \mathbb{Z}_{\ge 0} \right\}
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xn=zero*⁡ x∈Sψ ⁣(x)   where S={(0,∞),n=0(−n,−n+1),n<0x_{n} = \mathop{\operatorname{zero*}\,}\limits_{x \in S} \psi\!\left(x\right)\; \text{ where } S = \begin{cases} \left(0, \infty\right), & n = 0\\\left(-n, -n + 1\right), & n < 0\\ \end{cases}
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xn∼−n+1πatan⁡ ⁣(πlog⁡(n)),  n→∞x_{n} \sim -n + \frac{1}{\pi} \operatorname{atan}\!\left(\frac{\pi}{\log(n)}\right), \; n \to \infty

Derivatives and differential equations

Related topic: Gamma function
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ψ ⁣(z)=Γ′(z)Γ(z)\psi\!\left(z\right) = \frac{\Gamma'(z)}{\Gamma(z)}
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ψ ⁣(z)=ddz [log⁡Γ(z)]\psi\!\left(z\right) = \frac{d}{d z}\, \left[\log \Gamma(z)\right]
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ψ(m) ⁣(z)=dm+1dzm+1[log⁡Γ(z)]\psi^{(m)}\!\left(z\right) = \frac{d^{m + 1}}{{d z}^{m + 1}} \left[\log \Gamma(z)\right]
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dndzn[ψ ⁣(z)]=ψ(n) ⁣(z)\frac{d^{n}}{{d z}^{n}} \left[\psi\!\left(z\right)\right] = \psi^{(n)}\!\left(z\right)
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dndzn[ψ(m) ⁣(z)]=ψ(m+n) ⁣(z)\frac{d^{n}}{{d z}^{n}} \left[\psi^{(m)}\!\left(z\right)\right] = \psi^{(m + n)}\!\left(z\right)

Series representations

Series of rational functions

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ψ ⁣(z)=−γ+∑n=0∞(1n+1−1n+z)\psi\!\left(z\right) = -\gamma + \sum_{n=0}^{\infty} \left(\frac{1}{n + 1} - \frac{1}{n + z}\right)
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ψ(m) ⁣(z)=(−1)m+1m!∑n=0∞1(n+z)m+1\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \sum_{n=0}^{\infty} \frac{1}{{\left(n + z\right)}^{m + 1}}

Taylor series

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ψ ⁣(1+z)=−γ+∑n=1∞(−1)n+1ζ ⁣(n+1)zn\psi\!\left(1 + z\right) = -\gamma + \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} \zeta\!\left(n + 1\right) {z}^{n}

Laurent series

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ψ ⁣(z)=−1z−γ+∑n=1∞(−1)n+1ζ ⁣(n+1)zn\psi\!\left(z\right) = -\frac{1}{z} - \gamma + \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} \zeta\!\left(n + 1\right) {z}^{n}
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ψ ⁣(−n+z)=−1z+ψ ⁣(n+1)+∑k=1∞((−1)k+1ζ ⁣(k+1)+∑j=1n1jk+1)zk\psi\!\left(-n + z\right) = -\frac{1}{z} + \psi\!\left(n + 1\right) + \sum_{k=1}^{\infty} \left({\left(-1\right)}^{k + 1} \zeta\!\left(k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + 1}}\right) {z}^{k}
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ψ(m) ⁣(−n+z)=(−1)m+1m!zm+1+∑k=0∞(k+1)m((−1)m+k+1ζ ⁣(m+k+1)+∑j=1n1jk+m+1)zk\psi^{(m)}\!\left(-n + z\right) = \frac{{\left(-1\right)}^{m + 1} m !}{{z}^{m + 1}} + \sum_{k=0}^{\infty} \left(k + 1\right)_{m} \left({\left(-1\right)}^{m + k + 1} \zeta\!\left(m + k + 1\right) + \sum_{j=1}^{n} \frac{1}{{j}^{k + m + 1}}\right) {z}^{k}

Asymptotic expansions

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ψ ⁣(z)=log⁡(z)−12z−∑n=1N−1B2n2nz2n+RN′(z)\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \sum_{n=1}^{N - 1} \frac{B_{2 n}}{2 n {z}^{2 n}} + R'_{N}(z)
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ψ(m) ⁣(z)=(−1)m+1m!(1mzm+12zm+1+∑n=1N−1(m+1)2n−1(2n)!B2nzm+2n)+RN(m+1)(z)\psi^{(m)}\!\left(z\right) = \frac{{\left(-1\right)}^{m + 1}}{m !} \left(\frac{1}{m {z}^{m}} + \frac{1}{2 {z}^{m + 1}} + \sum_{n=1}^{N - 1} \frac{\left(m + 1\right)_{2 n - 1}}{\left(2 n\right)!} \frac{B_{2 n}}{{z}^{m + 2 n}}\right) + {R}^{(m + 1)}_{N}(z)

Weierstrass product

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ψ ⁣(z)Γ(z)=−e2γz∏n=0∞(1−zxn)exp⁡ ⁣(zxn)\frac{\psi\!\left(z\right)}{\Gamma(z)} = -{e}^{2 \gamma z} \prod_{n=0}^{\infty} \left(1 - \frac{z}{x_{n}}\right) \exp\!\left(\frac{z}{x_{n}}\right)

Representation by other functions

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ψ(m) ⁣(z)=(−1)m+1m!ζ ⁣(m+1,z)\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, z\right)
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ψ(m) ⁣(z)=(−1)m+1m!Φ ⁣(1,m+1,z)\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \Phi\!\left(1, m + 1, z\right)
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ψ ⁣(z)=(z−1) 3F2 ⁣(1,1,2−z,2,2,1)−γ\psi\!\left(z\right) = \left(z - 1\right) \,{}_3F_2\!\left(1, 1, 2 - z, 2, 2, 1\right) - \gamma
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ψ ⁣(z)=−γ0 ⁣(z)\psi\!\left(z\right) = -\gamma_{0}\!\left(z\right)

Functional equations

Recurrence relations

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ψ ⁣(z+1)=ψ ⁣(z)+1z\psi\!\left(z + 1\right) = \psi\!\left(z\right) + \frac{1}{z}
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ψ ⁣(z+n)=ψ ⁣(z)+∑k=0n−11z+k\psi\!\left(z + n\right) = \psi\!\left(z\right) + \sum_{k=0}^{n - 1} \frac{1}{z + k}
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ψ ⁣(z−n)=ψ ⁣(z)−∑k=1n1z−k\psi\!\left(z - n\right) = \psi\!\left(z\right) - \sum_{k=1}^{n} \frac{1}{z - k}
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ψ(m) ⁣(z+1)=ψ(m) ⁣(z)+(−1)mm!zm+1\psi^{(m)}\!\left(z + 1\right) = \psi^{(m)}\!\left(z\right) + \frac{{\left(-1\right)}^{m} m !}{{z}^{m + 1}}
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ψ(m) ⁣(z+n)=ψ(m) ⁣(z)+(−1)mm!∑k=0n−11(z+k)m+1\psi^{(m)}\!\left(z + n\right) = \psi^{(m)}\!\left(z\right) + {\left(-1\right)}^{m} m ! \sum_{k=0}^{n - 1} \frac{1}{{\left(z + k\right)}^{m + 1}}
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ψ(m) ⁣(z−n)=ψ(m) ⁣(z)−(−1)mm!∑k=1n1(z−k)m+1\psi^{(m)}\!\left(z - n\right) = \psi^{(m)}\!\left(z\right) - {\left(-1\right)}^{m} m ! \sum_{k=1}^{n} \frac{1}{{\left(z - k\right)}^{m + 1}}

Reflection formula

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ψ ⁣(1−z)=ψ ⁣(z)+πcot⁡ ⁣(πz)\psi\!\left(1 - z\right) = \psi\!\left(z\right) + \pi \cot\!\left(\pi z\right)
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ψ(m) ⁣(1−z)=(−1)m(ψ(m) ⁣(z)+πdmdzmcot⁡ ⁣(πz))\psi^{(m)}\!\left(1 - z\right) = {\left(-1\right)}^{m} \left(\psi^{(m)}\!\left(z\right) + \pi \frac{d^{m}}{{d z}^{m}} \cot\!\left(\pi z\right)\right)

Multiplication theorem

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ψ ⁣(nz)=log⁡(n)+1n∑k=0n−1ψ ⁣(z+kn)\psi\!\left(n z\right) = \log(n) + \frac{1}{n} \sum_{k=0}^{n - 1} \psi\!\left(z + \frac{k}{n}\right)
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ψ(m) ⁣(nz)=1nm+1∑k=0n−1ψ(m) ⁣(z+kn)\psi^{(m)}\!\left(n z\right) = \frac{1}{{n}^{m + 1}} \sum_{k=0}^{n - 1} \psi^{(m)}\!\left(z + \frac{k}{n}\right)

Conjugate symmetry

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ψ ⁣(z‾)=ψ ⁣(z)‾\psi\!\left(\overline{z}\right) = \overline{\psi\!\left(z\right)}
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ψ(m) ⁣(z‾)=ψ(m) ⁣(z)‾\psi^{(m)}\!\left(\overline{z}\right) = \overline{\psi^{(m)}\!\left(z\right)}

Integral representations

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ψ ⁣(z)=∫0∞(e−tt−e−zt1−e−t) dt\psi\!\left(z\right) = \int_{0}^{\infty} \left(\frac{{e}^{-t}}{t} - \frac{{e}^{-z t}}{1 - {e}^{-t}}\right) \, dt
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ψ ⁣(z)=∫0∞(e−t−1(1+t)z)1t dt\psi\!\left(z\right) = \int_{0}^{\infty} \left({e}^{-t} - \frac{1}{{\left(1 + t\right)}^{z}}\right) \frac{1}{t} \, dt
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ψ ⁣(z)=−γ+∫011−tz−11−t dt\psi\!\left(z\right) = -\gamma + \int_{0}^{1} \frac{1 - {t}^{z - 1}}{1 - t} \, dt
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ψ ⁣(z)=−γ+∫0∞e−t−e−zt1−e−t dt\psi\!\left(z\right) = -\gamma + \int_{0}^{\infty} \frac{{e}^{-t} - {e}^{-z t}}{1 - {e}^{-t}} \, dt
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ψ ⁣(z)=log⁡(z)+∫0∞e−zt(1t−11−e−t) dt\psi\!\left(z\right) = \log(z) + \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{t} - \frac{1}{1 - {e}^{-t}}\right) \, dt
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ψ ⁣(z)=log⁡(z)−12z−2∫0∞t(t2+z2)(e2πt−1) dt\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - 2 \int_{0}^{\infty} \frac{t}{\left({t}^{2} + {z}^{2}\right) \left({e}^{2 \pi t} - 1\right)} \, dt
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ψ ⁣(z)=log⁡(z)−12z−∫0∞e−zt(12−1t+1et−1) dt\psi\!\left(z\right) = \log(z) - \frac{1}{2 z} - \int_{0}^{\infty} {e}^{-z t} \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{{e}^{t} - 1}\right) \, dt
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ψ(m) ⁣(z)=(−1)m+1∫0∞tme−zt1−e−t dt\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} \int_{0}^{\infty} \frac{{t}^{m} {e}^{-z t}}{1 - {e}^{-t}} \, dt
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ψ(m) ⁣(z)=−∫01tz−11−tlog⁡m ⁣(t) dt\psi^{(m)}\!\left(z\right) = -\int_{0}^{1} \frac{{t}^{z - 1}}{1 - t} \log^{m}\!\left(t\right) \, dt

Generating functions

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∑n=1∞ψ ⁣(n)zn=z(γ+log⁡ ⁣(1−z))z−1\sum_{n=1}^{\infty} \psi\!\left(n\right) {z}^{n} = \frac{z \left(\gamma + \log\!\left(1 - z\right)\right)}{z - 1}
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∑n=1∞ψ ⁣(n)znn!=z[dda  1F1 ⁣(a,2,z)]a=1−γ(ez−1)\sum_{n=1}^{\infty} \psi\!\left(n\right) \frac{{z}^{n}}{n !} = z \left[ \frac{d}{d a}\, \,{}_1F_1\!\left(a, 2, z\right) \right]_{a = 1} - \gamma \left({e}^{z} - 1\right)

Finite sums

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∑k=1nψ ⁣(k)=n(ψ ⁣(n+1)−1)\sum_{k=1}^{n} \psi\!\left(k\right) = n \left(\psi\!\left(n + 1\right) - 1\right)
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∑k=1nψ ⁣(kn)e2πrki/n=nlog⁡ ⁣(1−e2πri/n)\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) {e}^{2 \pi r k i / n} = n \log\!\left(1 - {e}^{2 \pi r i / n}\right)
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∑k=1nψ ⁣(kn)cos⁡ ⁣(2πrkn)=nlog⁡ ⁣(2sin⁡ ⁣(πrn))\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) \cos\!\left(\frac{2 \pi r k}{n}\right) = n \log\!\left(2 \sin\!\left(\frac{\pi r}{n}\right)\right)
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∑k=1nψ ⁣(kn)sin⁡ ⁣(2πrkn)=π(r−n2)\sum_{k=1}^{n} \psi\!\left(\frac{k}{n}\right) \sin\!\left(\frac{2 \pi r k}{n}\right) = \pi \left(r - \frac{n}{2}\right)

Bounds and inequalities

Real range and signs

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x∈(x0,∞)    ⟹    ψ ⁣(x)∈(0,∞)x \in \left(x_{0}, \infty\right) \;\implies\; \psi\!\left(x\right) \in \left(0, \infty\right)
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x∈(0,x0)    ⟹    ψ ⁣(x)∈(−∞,0)x \in \left(0, x_{0}\right) \;\implies\; \psi\!\left(x\right) \in \left(-\infty, 0\right)
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x∈(−n,xn)    ⟹    ψ ⁣(x)∈(−∞,0)x \in \left(-n, x_{n}\right) \;\implies\; \psi\!\left(x\right) \in \left(-\infty, 0\right)
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x∈(xn,−n+1)    ⟹    ψ ⁣(x)∈(0,∞)x \in \left(x_{n}, -n + 1\right) \;\implies\; \psi\!\left(x\right) \in \left(0, \infty\right)
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(x∈(0,∞)  and⁡  m∈Z≥1)    ⟹    ψ(m) ⁣(x)∈{(0,∞),m odd(−∞,0),m even\left(x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}_{\ge 1}\right) \;\implies\; \psi^{(m)}\!\left(x\right) \in \begin{cases} \left(0, \infty\right), & m \text{ odd}\\\left(-\infty, 0\right), & m \text{ even}\\ \end{cases}
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(m∈Z≥1  and⁡  m odd  and⁡  x∈R∖{0,−1,…})    ⟹    ψ(m) ⁣(x)∈(0,∞)\left(m \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; m \text{ odd} \;\mathbin{\operatorname{and}}\; x \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \;\implies\; \psi^{(m)}\!\left(x\right) \in \left(0, \infty\right)

Real upper and lower bounds

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ψ ⁣(x)<log⁡(x)\psi\!\left(x\right) < \log(x)
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ψ ⁣(x)<log⁡(x)−12x\psi\!\left(x\right) < \log(x) - \frac{1}{2 x}
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ψ ⁣(x)>log⁡(x)−1x\psi\!\left(x\right) > \log(x) - \frac{1}{x}
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(−1)m+1ψ(m) ⁣(x)<(m−1)!xm+m!xm+1{\left(-1\right)}^{m + 1} \psi^{(m)}\!\left(x\right) < \frac{\left(m - 1\right)!}{{x}^{m}} + \frac{m !}{{x}^{m + 1}}
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(−1)m+1ψ(m) ⁣(x)>(m−1)!xm+m!2xm+1{\left(-1\right)}^{m + 1} \psi^{(m)}\!\left(x\right) > \frac{\left(m - 1\right)!}{{x}^{m}} + \frac{m !}{2 {x}^{m + 1}}

Monotonicity

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ψ ⁣(x+y)>ψ ⁣(x)\psi\!\left(x + y\right) > \psi\!\left(x\right)
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{ψ(m) ⁣(x+y)>ψ ⁣(x),m evenψ(m) ⁣(x+y)<ψ ⁣(x),m odd\begin{cases} \psi^{(m)}\!\left(x + y\right) > \psi\!\left(x\right), & m \text{ even}\\\psi^{(m)}\!\left(x + y\right) < \psi\!\left(x\right), & m \text{ odd}\\ \end{cases}

Summation of rational functions

Infinite series

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∑n=0∞(−1)nn+a=12(ψ ⁣(a+12)−ψ ⁣(a2))\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n + a} = \frac{1}{2} \left(\psi\!\left(\frac{a + 1}{2}\right) - \psi\!\left(\frac{a}{2}\right)\right)
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∑n=0∞1(n+a)r=(−1)r(r−1)!ψ(r−1) ⁣(a)\sum_{n=0}^{\infty} \frac{1}{{\left(n + a\right)}^{r}} = \frac{{\left(-1\right)}^{r}}{\left(r - 1\right)!} \psi^{(r - 1)}\!\left(a\right)
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∑n=0∞(−1)n(n+a)r=(−1)r2r(r−1)!(ψ(r−1) ⁣(a2)−ψ(r−1) ⁣(a+12))\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{{\left(n + a\right)}^{r}} = \frac{{\left(-1\right)}^{r}}{{2}^{r} \left(r - 1\right)!} \left(\psi^{(r - 1)}\!\left(\frac{a}{2}\right) - \psi^{(r - 1)}\!\left(\frac{a + 1}{2}\right)\right)
21e21a
∑n=0∞1(n+a)(n+b)={1a−b(ψ ⁣(a)−ψ ⁣(b)),a≠bψ′ ⁣(a),a=b\sum_{n=0}^{\infty} \frac{1}{\left(n + a\right) \left(n + b\right)} = \begin{cases} \frac{1}{a - b} \left(\psi\!\left(a\right) - \psi\!\left(b\right)\right), & a \ne b\\\psi'\!\left(a\right), & a = b\\ \end{cases}
b7f13b
∑n=0∞(−1)n(n+a)(n+b)={12(a−b)(ψ ⁣(a2)+ψ ⁣(b+12)−ψ ⁣(b2)−ψ ⁣(a+12)),a≠b14(ψ′ ⁣(a2)−ψ′ ⁣(a+12)),a=b\sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{\left(n + a\right) \left(n + b\right)} = \begin{cases} \frac{1}{2 \left(a - b\right)} \left(\psi\!\left(\frac{a}{2}\right) + \psi\!\left(\frac{b + 1}{2}\right) - \psi\!\left(\frac{b}{2}\right) - \psi\!\left(\frac{a + 1}{2}\right)\right), & a \ne b\\\frac{1}{4} \left(\psi'\!\left(\frac{a}{2}\right) - \psi'\!\left(\frac{a + 1}{2}\right)\right), & a = b\\ \end{cases}

Finite sums

19e67f
∑n=AB1n+a=ψ ⁣(a+B+1)−ψ ⁣(a+A)\sum_{n=A}^{B} \frac{1}{n + a} = \psi\!\left(a + B + 1\right) - \psi\!\left(a + A\right)

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

2021-03-15 19:12:00.328586 UTC