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.

Filter by
Sorted by
Tagged with
5 votes
2 answers
340 views

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)_{...
Assinisa Hamidata's user avatar
1 vote
0 answers
35 views

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 $n,a,b$ ...
Rekha K.'s user avatar
1 vote
0 answers
84 views

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 (...
Mir Aaliya's user avatar
1 vote
1 answer
161 views

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 ...
Manfred Weis's user avatar
  • 12.6k
0 votes
0 answers
352 views

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.
Yaakov Baruch's user avatar
6 votes
2 answers
402 views

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. ...
forwhole's user avatar
6 votes
2 answers
259 views

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)\...
Julian's user avatar
  • 613
1 vote
0 answers
115 views

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)...
Diego Santos's user avatar
5 votes
1 answer
739 views

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....
jtkw's user avatar
  • 63
2 votes
0 answers
117 views

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, ...
Mr.MathDoctor's user avatar
1 vote
1 answer
243 views

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 ...
Guido Li's user avatar
0 votes
0 answers
52 views

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 ...
Jojo's user avatar
  • 333
3 votes
0 answers
175 views

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 ...
183orbco3's user avatar
  • 271
0 votes
0 answers
103 views

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}...
Eduardo's user avatar
2 votes
1 answer
173 views

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 $$\...
Mr.MathDoctor's user avatar
0 votes
0 answers
72 views

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 ...
NuKuYul's user avatar
  • 71
1 vote
0 answers
77 views

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 ...
Lateef's user avatar
  • 91
2 votes
1 answer
74 views

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 $P_i(z_j)=\delta_{ij}$?
ABB's user avatar
  • 3,898
5 votes
2 answers
396 views

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 ...
xiangsha's user avatar
6 votes
1 answer
422 views

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 ...
Joe's user avatar
  • 535
1 vote
0 answers
63 views

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/...
JZS's user avatar
  • 469
3 votes
0 answers
119 views

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 ...
Bytegear's user avatar
  • 123
5 votes
0 answers
191 views

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} = \...
mathoverflowUser's user avatar
4 votes
1 answer
415 views

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&...
Iceman's user avatar
  • 43
1 vote
0 answers
204 views

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 ...
Laithy's user avatar
  • 865
1 vote
1 answer
101 views

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))...
Tyma Gaidash's user avatar
7 votes
0 answers
436 views

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 ...
Max Muller's user avatar
  • 4,485
3 votes
0 answers
191 views

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(...
Martin's user avatar
  • 1,101
3 votes
0 answers
242 views

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,\...
Michael Hardy's user avatar
0 votes
1 answer
98 views

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 $ $f_i(x,0)=x_i$ $\...
sorrymaker's user avatar
2 votes
1 answer
318 views

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 ...
Diego Santos's user avatar
1 vote
0 answers
66 views

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 \\&...
Maxim Lyutikov's user avatar
0 votes
2 answers
212 views

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 ...
Student's user avatar
  • 653
2 votes
1 answer
131 views

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 ...
Student's user avatar
  • 653
8 votes
3 answers
1k views

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....
J.Ham's user avatar
  • 83
1 vote
1 answer
102 views

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 $x \in \mathbb{R}$.}...
April's user avatar
  • 389
1 vote
0 answers
60 views

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)...
ad ab's user avatar
  • 11
3 votes
2 answers
270 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{...
Z. Alfata's user avatar
  • 640
0 votes
1 answer
103 views

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/
user338431's user avatar
2 votes
1 answer
327 views

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 $0<Re(z)<1$;...
user363337's user avatar
2 votes
2 answers
268 views

Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere

This question is related to: https://math.stackexchange.com/q/4270522/168758 Let $H_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the ...
dohmatob's user avatar
  • 6,716
5 votes
0 answers
220 views

Is there a special function for $\sum_{m=0}^{\infty} x^m/(m!)^s$?

Is there a special function for the following series? $$\sum_{m=0}^{\infty} {x^m \over (m!)^s}$$ Here, $s$ is a positive real number. When, $s$ is an integer, $s=n \in \mathbb{Z}$, this series can be ...
TheTwistedSector's user avatar
4 votes
1 answer
438 views

Proof identity for hypergeometric series 2F1(a,b;c;x)

I would like prove the following identity: $$ _2F_1(a,b;c,x) = \frac{c+(a−b+1)x}{c} {} _2F_1(a+1,b;c+1;x) - \frac{(a+1)(c−b+1)x}{c(c+1)} {} _2F_1(a+2,b;c+2;x). $$ I've tried this so far: I know that $...
Mathstudent's user avatar
1 vote
2 answers
490 views

Recurrence for the sum

Let $m\geq 2$ be a fixed integer. Let $$f(n):=\begin{cases} mf\left(\frac{n}{m}\right),&\text{if $n\mod m = 0$;}\\ 1,&\text{otherwise} \end{cases}$$ then if we have $$a(n):=\begin{cases} 1,&...
Notamathematician's user avatar
2 votes
1 answer
504 views

What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...
qifeng618's user avatar
  • 836
1 vote
0 answers
89 views

Recurrence for the viabin numbers of the self-conjugate integer partitions

Let $a(n)$ be A290254, the viabin numbers of the self-conjugate integer partitions, also defined as $\left\lbrace 0 \right\rbrace$ union fixed points of A059894, self-inverse permutation defined as ...
Notamathematician's user avatar
1 vote
0 answers
117 views

Identities on the Whittaker function $W_{-\kappa,\mu}(z)$?

As in (for example) [Magnus, W., Oberhettinger, F., Soni, R.: Formulas and Theorems for the Special Functions of Mathematical Physics, Grundlehren der mathematischen Wissenschaften 52, Springer, ...
Z. Alfata's user avatar
  • 640
2 votes
0 answers
89 views

A subsequence expressed in terms of a sum with a triangle

We have a sequence which generalize A329369: $$a(2n+1, p, q) = a(n, p,q), a(2n, p , q) = pa(n, p,q) + qa(n - 2^{f(n)}, p,q) + a(2n - 2^{f(n)}, p,q), a(0, p, q) = 1$$ where $f(n)$ is A007814, exponent ...
Notamathematician's user avatar
1 vote
1 answer
215 views

A determinant involving the cotangent function

Let $n>1$ be odd. In my 2019 preprint On some determinants involving the tangent function, I proved that $$\det\left[\tan\pi\frac{aj+bk}n\right]_{1\le j,k\le n-1}=\left(\frac{-ab}n\right)n^{n-2}$$ ...
Zhi-Wei Sun's user avatar
  • 14.4k
11 votes
1 answer
903 views

How to determine if you've discovered a new identity for a special function

Often times, we consult resources, like Abramowitz and Stegun's Handbook of Mathematical Functions https://www.math.ubc.ca/~cbm/aands/, NIST's database on special functions https://www.nist.gov/...
garserdt216's user avatar

1 2
3
4 5
17