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

Digamma function