# Fungrim entry: 88e89f

$\sum_{n=1}^{\infty} \psi\!\left(n\right) \frac{{z}^{n}}{n !} = z \left[ \frac{d}{d a}\, \,{}_1F_1\!\left(a, 2, z\right) \right]_{a = 1} - \gamma \left({e}^{z} - 1\right)$
Assumptions:$z \in \mathbb{C}$
TeX:
\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}
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
DigammaFunction$\psi\!\left(z\right)$ Digamma function
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
Infinity$\infty$ Positive infinity
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Hypergeometric1F1$\,{}_1F_1\!\left(a, b, z\right)$ Kummer confluent hypergeometric function
ConstGamma$\gamma$ The constant gamma (0.577...)
Exp${e}^{z}$ Exponential function
CC$\mathbb{C}$ Complex numbers
Source code for this entry:
Entry(ID("88e89f"),
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))))),
Variables(z),
Assumptions(Element(z, CC)))

## Topics using this entry

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