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Fungrim entry: 50f57e

n=0(2nn)xnn!=e2xI0 ⁣(2x)\sum_{n=0}^{\infty} {2 n \choose n} \frac{{x}^{n}}{n !} = {e}^{2 x} I_{0}\!\left(2 x\right)
Assumptions:xCx \in \mathbb{C}
TeX:
\sum_{n=0}^{\infty} {2 n \choose n} \frac{{x}^{n}}{n !} = {e}^{2 x} I_{0}\!\left(2 x\right)

x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
Factorialn!n ! Factorial
Infinity\infty Positive infinity
Expez{e}^{z} Exponential function
BesselIIν ⁣(z)I_{\nu}\!\left(z\right) Modified Bessel function of the first kind
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("50f57e"),
    Formula(Equal(Sum(Mul(Binomial(Mul(2, n), n), Div(Pow(x, n), Factorial(n))), Tuple(n, 0, Infinity)), Mul(Exp(Mul(2, x)), BesselI(0, Mul(2, x))))),
    Variables(x),
    Assumptions(Element(x, CC)))

Topics using this entry

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

2019-08-21 11:44:15.926409 UTC