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Fungrim entry: 88e89f

n=1ψ ⁣(n)znn!=z[dda1F1 ⁣(a,2,z)]a=1γ(ez1)\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)
Assumptions:zCz \in \mathbb{C}
\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)

z \in \mathbb{C}
Fungrim symbol Notation Short description
Sumnf(n)\sum_{n} f(n) Sum
DigammaFunctionψ ⁣(z)\psi\!\left(z\right) Digamma function
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Hypergeometric1F11F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric function
ConstGammaγ\gamma The constant gamma (0.577...)
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
Source code for this entry:
    Formula(Equal(Sum(Mul(DigammaFunction(n), Div(Pow(z, n), Factorial(n))), For(n, 1, Infinity)), Sub(Mul(z, ComplexDerivative(Hypergeometric1F1(a, 2, z), For(a, 1))), Mul(ConstGamma, Sub(Exp(z), 1))))),
    Assumptions(Element(z, CC)))

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2021-03-15 19:12:00.328586 UTC