Questions tagged [special-functions]

Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.

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What's the convergence condition for the generating function formula of Legendre polynomials?

What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$? $$ \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n $$ I know it is convergent at least ...
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Can a general quintic be solved using Inverse Beta Regularized function?

Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
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Is it possible to solve sextic equations using the Fox H function?

Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago. In contrast, we know more about the Fox H function, and we ...
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Generalization of identity for terminating hypergeometric function

Let ${}_2F_1(a,b;c;z)$ be the ordinary hypergeometric function for $z \in \mathbb{C}$ \begin{equation} {}_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{z^k}{k!} \frac{(a)_{k} (b)_k}{(c)_k}\,, \end{...
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What's the fastest way to compute $\log n$ for $n>1$?

As it is well known, if $|x|<1$ then we can compute $\log(1+x)$ by the Taylor series $$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots.$$ Thus, to compute $\log n$ with $n>1$, we may employ the ...
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Zeros of hypergeometric functions with complex variables

Let $z$ be a complex number and let $a,b,c > 0$. I would like to know the zeros of the following hypergeometric function: $$_{2}F_{1} (a,b; c :z )=\sum_{k=0}^{+ \infty} \frac{(a)_{k}(b)_{k}}{(c)_{...
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How to relate this integration with the integral expansion of special functions?

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
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Is this function concave? If it is, can we show it in a theoretical way?

Suppose we have a function: \begin{equation} \begin{aligned} f(x_1,x_2,\cdots,x_n)&=\sum\limits_{i=0}^n\frac{(x_i e^{-x_i}-x_{i+1}e^{-x_{i+1}})^2}{e^{-x_i}-e^{-x_{i+1}}}\\ &=\frac{(x_0 e^{-x_0}...
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What is t-equivalence in function spaces?

In $C_p$-Theory monographs, it is said that two topological spaces $X$ and $Y$ are said to be $t$-equivalent means that $C_p(X)$ is homeomorphic to $C_p(Y)$. Then they also define $u$-equivalences (...
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Find $x$ that solves $x\left(e^{\frac{a}{x}}-1\right)-y=0$

When trying to solve the equation in the title with WA, it produced the following as the solution: now, if you divide the numerator and denominator by $y$ and set $z:=-\frac{a}{y}$ the solution ...
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Is $\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi$?

It seems that $$\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi.$$ But I can't prove it. I cannot prove that the function is decreasing in $x$ either.
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Cardinality of a special set of continuous functions

Let $C$ be a set of continuous functions with a domain $[0,1]$ and for every input $x$ in a domain there is a set $S(x)$ that contains all values that functions in $C$ will output given that input. ...
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Appearances of the basic hypergeometric series ${}_0\phi_1(;z;q;q^l z)$

On wikipedia one can find the general definition of a unilateral basic hypergeometric series ${}_r\phi_s$. The special case ${}_0\phi_1(;z;q;q^l z)$ has the expansion $$ {}_0\phi_1(;z;q;q^l z) = \sum_{...
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Asymptotics of error function integral with square root

I am interested in the asymptotics of the integral $$I(a):=\int_0^\infty \sqrt{x}\operatorname{Erfc}(x+a)\,\mathrm{d}x$$ for $a>0$. I think that $I(a)$ should be decaying exponentially as $I(a)\...
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About writing solutions of linear ODE's: Is this statement correct?

A motivating example: Consider the Hypergeometric equation $$z(1-z) \frac{d^2y}{dz^2}+(c-(a+b+1)z) \frac{dy}{dz}-aby=0,$$ it has a solution given by the Gauss's Hypergeometric function $$_2F_1(a,b;c;z)...
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Any name for this special function?

We know $$ \sum_{m=0}^\infty \frac{x^m}{(a-m)!m!} = \frac{1}{a!}(1+x)^m $$ where we understand the factorial as Gamma function $\Gamma(x)$ such that it is divergent if the argument is negative integer....
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Motivation behind the Bohr-Mollerup Theorem relating the Gamma function

In Wikipedia, it states about the Bohr-Mollerup Theorem: The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved. If anyone knows, ...
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Exponential decay bound on integral

I have an integral of the form $$ \int_R^{\infty} e^{-x} x^n \vert L_m^{\alpha}(x) \vert^2 \ dx,$$ where $L_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$ I would to get a nice ...
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Integral involving the product of Kummer Confluent Hypergeometric and exponential functions

I am having trouble with calculating the following integral: $$\int_{0}^{z} x^{b - 1} (z - x)^{\alpha - 1} e^{-\beta x} {}_1 F_1{(a;b;x)} \, dx,$$ where ${}_1F_1$ is the Kummer Confluent ...
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How to know if two special functions are related by an elementary function?

Suppose I have two special functions $f_1$ and $f_2$. Is there an algorithm which can tell me whether there exists elementary $g$ such that $f_1 = g\circ f_2$? Furthermore, is there any possibility to ...
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Is factorial the restriction of some elementary function?

Hölder's theorem says that Gamma function is very non-elementary, but it does not exclude the possibility that factorial is the restriction of some elementary function to natural numbers. The answer ...
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Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$

I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral: $$\frac{1}{\pi}...
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Bounds for ratio of Bessel functions

I know several good papers where bounds of Bessel function ration considered. For instance, the following Bessel function ratio, $$h(z) = \frac{I_{\nu}(z)}{I_{\nu+1}(z)}$$ has bounds $$h(z)<\frac{z}...
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2 votes
1 answer
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Gamma function and the somewhat extended version of Bohr-Mollerup theorem

The Gamma function $\Gamma$ is defined by \begin{equation*} \Gamma(x)=\int_{0}^\infty t^{x-1}e^{-t} \,\mathrm{d}t, \end{equation*} for $x>0$. It satisfies the well-known functional equation $$\...
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Closed form of integral $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$

How could one calculate the closed form solution of this integral: $\mathcal{P}\int_{-\infty}^{+\infty}dy\frac{ \text{Li}_{k}(\exp(-(y-\frac{\xi }{2})^2))}{\exp (2 \xi y)-1}$ Here the integral is ...
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Real life applications of distributions through models or simulations [closed]

What are the areas we can apply distributions in classical harmonic analysis? I don't mean probability distributions but distributions that are continuous linear functionals on the space of test ...
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Two variable polynomials that behave like Lagrange polynomials [closed]

Let us consider different points $z_i=(x_i,y_i)$ in the plane where $i=1,\cdots n$. Q Do there exist two variable polynomials $P_i(x,y)$ with minimal degree such that $P_i(z_j)=\delta_{ij}$?
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How to integrate the multinomial over a ball in $\mathbb{R}^{n}$?

I got an interesting question. Consider this integral: $$ \int_{B(0,1)}\bigg(\sum_{j=1}^{n}a_{j}x_{j}^2\bigg)^m \mbox{d}x, \quad m,n\in \mathbb{N}, \ a_{i}>0, \ i=1,2,\ldots,n.$$ It is clear that ...
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Subtraction of two similar Meijer G-functions

Can we get another Meijer G-function from the following subtraction? $$ G^{0, 5}_{5, 2}\left( z \left| \begin{matrix} (1, a, b, c, d) \\ (e, f) \end{matrix} \right.\right) -G^{0, 5}_{5, 2}\left( z \...
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Diagonalizing the ‘restricted’ Hilbert transform on $L^2(0,1)$, $f(z_1) \mapsto \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2$

Consider the following operator on functions $\mathcal{T}: L^2(0,1) \to L^2(0,1)$ over the complex numbers. \begin{equation} (\mathcal{T} f)(z_1) = \mathrm{p.v.} \int_0^1 \frac{i}{z_1-z_2}f(z_2) dz_2 ...
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Show that an integral operator with Bessel function kernel is bounded on $L^2(0,\infty)$

Let $J_0$ denote the Bessel function of the first kind of order $\nu = 0$ (see DLMF 10.2), $$ J_0(z) = \sum_{k = 0}^\infty (-1)^k \frac{(\tfrac{1}{4} z^2)^k }{k! \Gamma(k + 1)}. $$ Put $u_0(r) = r^{1/...
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Question on the model completeness of the real field expanded by restricted Pfaffian functions

Currently I'm reading "Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function" by Wilkie, and I do not ...
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An addition theorem for three functions similar to $\sin,\cos$ and $\sinh,\cosh$ and one / some questions?

Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy: $ \begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} = \...
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Speed of convergence of $\zeta(2k)\to 1$?

From the definition of $\zeta(z):= \sum_{k=1}^\infty \tfrac{1}{k^z}$ for $\mathrm{Re}(z)>1$ it is obvious that $\zeta(2k)\downarrow 1$ as $k \rightarrow \infty$. I am interested in the "true&...
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Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics

Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$? One ...
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Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function

Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series $$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
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Is there any literature on $\sum_{i=1}^{k} \left[ {k \atop i} \right] \zeta(i+1) $?

As per these questions, I'm trying to evaluate $$\sum_{n=2}^{\infty} \big{(} \zeta(n)^{2}-1 \big{)} = 1+ \sum_{m=2}^{\infty} \frac{H_{-\frac{1}{m}}}{m}. $$ Here, $H_{x}$ is a generalized Harmonic ...
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Can $\exp(W(\sqrt{\ln(\sqrt{n})})$ be an integer?

Let $W(z)$ be the Lambert $W$ function and $n$ a positive integer. Is it possible that $\exp(W(\sqrt{\ln(\sqrt{n})})$ is an integer? If $\exp(W(z))$ is an integer, say $k$, then we get $\frac{z}{k}=W(...
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Trigonometry and plane geometry

This will be a variation on the theme of this question, or maybe a rephrasing of it with a somewhat readjusted emphasis. In this posting I introduced the function \begin{align} & f_3(\theta_1,\...
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Looking for some special functions

I'm looking for some continuous functions $\{f_i(x,t)\}$, here $x=(x_1,x_2..., x_n)$, such that: $f_i(x,t):R^n\times [0, +\infty)\rightarrow R ~~\text{is continuous for each}~~i $ $f_i(x,0)=x_i$ $\...
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2 votes
1 answer
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What functions do we need to solve linear second order differential equations with polynomial coeficients? [closed]

. Final edit: The problem I had in mind is properly asked in THIS MO QUESTION, so I'll vote to close the present post e recommend anyone interested in the topic to visit that link. . . . . . Below is ...
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1 vote
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Airy-type integrals (with different power $\neq 3$)

I am looking for integrals closely related to the Airy function \begin{eqnarray} && A_1= \int _0^\infty x \sin \Phi dx \nonumber \\ && A_2= \int _0^\infty \cos \Phi dx \nonumber \\&...
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2 answers
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Asymptotic for eigenvalues for the following ode?

Consider the following Sturm-Liouville problem, $$(\sqrt{\sin \theta} Y')' + \lambda \sqrt{\sin \theta} Y =0$$ where $Y(\theta):[0,\pi] \to \mathbb{R}$ with boundary conditions $Y'(0)=Y'(\pi)=0.$ I ...
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2 votes
1 answer
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How to compute this limit involving the associated Legendre function?

I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the ...
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4 votes
2 answers
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Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?

This seems such a simple question that I fear I must have missed some elementary maths. I am looking for a way to solve $x+x^a = y$ by reference to an already defined function, $a,x,y > 0$ are real....
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1 vote
1 answer
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Comparing different norms of a polynomial

For $m\in \mathbb{N}$ and $a=(a_0,a_1,\ldots,a_{m}) \in \mathbb{R}^{m+1}$, consider the polynomial $P_{a}$ defined by $$ P_{a} (x):= a_0 + a_1 x^2 + \ldots + a_{m}x^{2m}\text{, for $x \in \mathbb{R}$.}...
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Integrability of an alternating series with hypergeometric coefficients

during my research, I came up with the following series $$ f(t):=\sum_{k=0}^\infty \frac{\left(-t^2\right)^k}{(k!)^2}{}_3F_{2}\left(\left(-k,-k,-k\right);\left(1,\frac{1}{2}-k\right);\frac{1}{4}\right)...
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3 votes
2 answers
214 views

$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function

I try to calculate the following series \begin{align*} S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k}, \end{...
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Hyper geometric series reference

Can someone point out a reference for the proof of this identity? Thanks in advance. https://functions.wolfram.com/HypergeometricFunctions/Hypergeometric4F3/03/03/01/0002/
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2 votes
1 answer
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Upper bound for the complex Beta function

The question is almost the same as here. What is the upper bound for a complex Beta function $\displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)}$ with $0<Re(s)<1$ and $0<Re(z)<1$;...
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