Questions tagged [special-functions]

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

3,451 questions
40 views
+50

881 views

Ramanujan Summation

It seems that under the light of Ramanujan Summation the following is plausible: $$1 + {2^{2n - 1}} + {3^{2n - 1}} + \cdots = - \frac{{{B_{2n}}}}{{2n}}(\Re)$$ Alas, I can't really find any ...
161 views

Hunt for exact solutions of second order ordinary differential equations with varying coefficients.

Let $a,a_1,a_2,b \in {\mathbb R}$. Being inspired by the answer to Solve $y''(x)=[a(x^2-1)^2+b]y(x)$ we found solutions of the following second order ODE : \begin{equation} \frac{d^2 y(x)}{d ...
137 views

Hypergeometric series for $\mathrm{Cl}_2(\pi/3)$

I am trying to find a hypergeometric series for $\mathrm{Cl}_2(\pi/3)$, where $$\mathrm{Cl}_2(x)=-\int_0^x\log\left|2\sin\frac{t}2\right|dt=\sum_{k\geq1}\frac{\sin kx}{k^2}$$ Is the Clausen ...
1k views

11 views

Relation between two Fox-H function with positive and negative argument

Is there the relation between Fox-H function with positive argument and Fox-H function with negative argument? My question is attached in the image.
15 views

Relation for two the Fox-H function with positive and negative argument

Is there the relation between Fox-H function with positive argument and Fox-H function with negative argument? My question is attached in the image.
1k views

A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
39 views

162 views

Solution of $f(x)^2\frac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
40 views

Series with gamma functions

I would like to understand how can I write down the expression for the following series: $$S_0=\sum_{k=2}^{\infty}(-1)^kA^{k}\frac{\Gamma(k-3/2)}{\Gamma(k+1)}.$$ I have seen related topics on this ...
20 views

A System Involving the Trigamma Function

Given $K \geq 2$ real numbers $a_1, \dots, a_K$, with $a_k > 0$ for $k=1,\dots,K$, consider the system of equations \begin{equation} (a_k - x_k) \psi^{(1)}(x_k) = \psi^{(1)} \left(\sum_{k=1}^{K}...
700 views