# Questions tagged [special-functions]

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

1,001 questions
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### Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
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### Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
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### Solution to Vector Lambert W function type Equation

I was wondering if anyone has any ideas for a closed-form solution to the equation $$Ax + \exp(x) +b =0$$ where $x,b \in \mathbb{R}^n$, $A$ is a symmetric positive definite matrix and $\exp$ denotes ...
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### Is there a closed form for $\,_4 F_3(1,1,1,3; 3/2,5/2,5/2;1)$?

A semi-algebraic generalization of the Steiner surface has appeared, $$S = \left\{(x,y,z,t) \space \vert \space t^2(1-x^2-y^2-z^2-t^2) - (x^2 y^2 + x^2 z^2 + y^2 z^2 - 2 x y z) \geq 0 \right\}$$ ...
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### Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) ,$$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the ...
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### Integral of a product of five Bessel functions of order $0$

Does the following integral have a closed form? $$\mathcal{J}(2,3,5,7,11) = \int_0^\infty x J_0(x\sqrt{2})J_0(x\sqrt{3})J_0(x\sqrt{5})J_0(x\sqrt{7})J_0(x\sqrt{11})\,dx.$$ I know that some similar ...
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The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
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### Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: $$I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt,$$ \...
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### relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions

There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
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### More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

In this post, the OP asks about the integral, $$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$ I. User DavidH gave a beautiful (albeit long)...
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### The choice of contour in the definition of Meijer G

It appears that when the Meijer G function is discontinuous on the unit circle, the integrals over the left and the right loops can exist but differ. For \$G_{1,1}^{0,1}\left(z\,\middle|\begin{array}{c}...