# 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.

797
questions

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### Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?

I. Two recurrence relations
The first one was also discussed in this MO post. We have the similar,
\begin{align}
(n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...

5
votes

1
answer

205
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### On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...

-2
votes

1
answer

135
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### Two-variable continuous function which results in an integer if and only if arguments are integer

I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
$f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
$f(m,n) \le f(...

2
votes

2
answers

174
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### Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$

(Note: This third method continues from this post.)
There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...

10
votes

1
answer

377
views

### On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$?

I. Four quintics?
The general quintic can be transformed in radicals to at least three one-parameter forms. For simplicity, assume this free parameter to be some generic "alpha". Hence,
$$x^...

4
votes

2
answers

456
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### Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$

I. Degree 8
Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$
\begin{align}
{j_1}\; &=\frac{(x^2 + ...

-1
votes

0
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33
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### Finding a double or higher order sum of Lorentzian function products centered at integer grid points. Can this be approximated by a Riemann integral?

I am trying to evaluate this sum over a large number of N $\sum_{m1 = 0}^N \sum_{m2 = 0}^N \sum_{m3 = 0}^N \frac{1}{[w1 - (m1 + m2)(k/N) + i g1] [w2 - (m1 - m3)(k/N) + i g2] [w3 - (m1 + m2-m3)(k/N) + ...

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votes

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answers

33
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### Functions that have their second derivatives as a lipschitz function [migrated]

What would be an example of a function that is not $C^3$ on $[0,1]$, but has its second derivative as a Lipschitz function?

2
votes

0
answers

41
views

### Transforming a Fuchsian equation with four finite singularities to the Heun equation

I keep seeing it claimed that the general second-order Fuchsian equation with four singularities can be transformed to the Heun equation, but I have never seen anyone explicitly write out the steps, ...

2
votes

0
answers

141
views

### What is known about "anti polynomials"?

I recently encountered a problem whose solution required solving $f(x):=\sum\alpha_i r_i^x\, =\, c;\ \alpha_i,r_i,x\in\mathbb{R},i\in I\subset\mathbb{N}, $ for $x$.
While the Newton method solves the ...

1
vote

0
answers

119
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### Ask for a proof of an inequality involving the Bernoulli numbers

Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...

4
votes

1
answer

334
views

### Inequality for Fourier transform of a power exponential function

Let
$$
f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }},
x \in \mathbb{R}, 0<\alpha<2,
$$
where
$\phi_1(\alpha)=\frac{\alpha}{2}
\left\{{\{\Gamma(3/\alpha)\}^{1/...

4
votes

4
answers

494
views

### What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?

It is known that
\begin{equation*}
\tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation*}
and
\begin{equation*}
\ln\tan x=\ln x+\...

2
votes

0
answers

95
views

### Asymptotic expansion of Jacobi function

For $\alpha,\beta \in \mathbb{C},\, \alpha$ a non-negative integer, we define $$A_{\alpha,\beta}(t)=(\sinh t)^{2\alpha+1}(\cosh t)^{2\beta+1} $$
and $$ \mathcal{L}_{\alpha,\beta}=\frac{d^2}{dt^2}+\...

4
votes

0
answers

120
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### Asymptotic analysis for a double integral related to Airy functions

Let $Ai(x,y)$ be the Airy kernel which is given by
\begin{equation}\label{equ2.12}
Ai(x,y)=
\begin{cases}
\dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\
Ai'(x)^2-xAi(x)^2 & x=y. \\
\end{...

1
vote

0
answers

96
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### Infinite series involving generalised hypergeometric functions

I've recently stumbled into hypergeometric functions while trying to evaluate the integral:
$$
\int \exp \big( x^2 + bx + c \big) {\rm erf} ( x ) \operatorname{d\!}x
$$
Essentially, working from an ...

0
votes

0
answers

88
views

### The upper bound of hyperbolic cosine function in complex plane

I want to find the upper bound of the following function :
\begin{equation}
M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right|
\end{equation}
where
\begin{equation}
\begin{aligned}
& 4 \...

0
votes

0
answers

65
views

### Expanding 3F2 functions in terms of polylogarithms

I need to expand hypergeometric functions in the form of ${}_3F_2(1, 1, k; m, n)$ and ${}_3F_2(1, 1, 1; m, n)$ with $k < m \le n$ and $k, m, n \in \mathbb{Z}$, in terms of polylogarithms.
The ...

3
votes

0
answers

333
views

### Does the integral $\int_0^{\infty}e^{cx^2+dx}dx/(a+bx)$ have a closed form?

The integral is
$$
\int_0^{\infty}\frac{e^{-cx^2+dx}}{a+bx}dx.
$$
Here I assume that $a,b,c,d$ are chosen to make this integral convergent. Rewritting the rational function as a integral over ...

0
votes

1
answer

271
views

### What are the properties of umbra with moments $\{1,1/2,1/3,1/4,1/5,...\}$?

If we apply operator $D\Delta^{-1}$ to a function, we will get the (Bernoulli) umbral analog of the function. Particularly, applying it to $x^n$ we will get the Bernoulli polynomials $B_n(x)$. ...

3
votes

1
answer

220
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### Integrating $\int_0^\infty \sqrt x e^{-4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$

Show that
$$I= \int_0^\infty \sqrt x e^{\large -4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$$
$$=\frac{1}{3}-\frac{\sqrt[3]{2\sqrt 3+3}+\sqrt[3]{2\sqrt 3-3}...

5
votes

2
answers

287
views

### Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$?

By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$):
$$J_0(u)J_0(v)+2\sum_{n=1}^\infty J_n(u)J_n(v) \cos(n\alpha) = J_0 \left( \sqrt{u^2+v^2-2uv \...

0
votes

1
answer

193
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### Identity involving Stirling number of the second kind

I'm looking for a citable reference for the following identity involving the Stirling numbers of the second kind $S(n, k)$ stated in Equation (27): For $n \geq 2$,
$$
\sum_{m=1}^n S(n, m) (-1)^m (m-1)!...

2
votes

0
answers

96
views

### Power series of the modified Bessel function of the second kind

I am looking for a power series representation of
$$ \frac{1}{K_{\nu}(x)}, $$
where $K_{\nu}$ denotes the modified Bessel function of the second kind and $\nu>-1/2$ is not an integer.
I know that ...

0
votes

1
answer

85
views

### Solutions of complex linear difference equations

I'm wondering what the solutions of complex linear difference equations like
\begin{equation}
f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C}
\end{equation}
look ...

2
votes

1
answer

134
views

### Asymptotic analysis of an expression involving a Fox's H function

One of the performance metrics calculated in the analysis of telecommunications systems is the ergodic channel capacity, $C_{\rm erg}$. During one of my studies, I found the expression below for such ...

0
votes

0
answers

51
views

### Fourier transform of an exponential function with radical argument divided by a radical

I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...

10
votes

1
answer

572
views

### Rigorous proof of the pentagon identity

I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.
For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...

1
vote

0
answers

123
views

### What pre knowledge does Mumford's Tata collections on theta need?

I am a sophomore and have the foundation of complex analysis and abstract algebra. I have learned Legendre elliptic integral and Jacobian elliptic function, and it is through this that I know theta ...

1
vote

1
answer

241
views

### Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential ...

2
votes

1
answer

122
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### Ask for references or proofs of two explicit formulas for special Gauss hypergeometric functions

Can one supply related references or detailed proofs of the following two explicit formulas?
$$
{}_2F_1\biggl(2\alpha+1,2;\alpha+3;\frac{1}{2}\biggr)
=2\frac{\alpha B(1/2,\alpha)-1-\alpha}{(1-\alpha) ...

4
votes

1
answer

110
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### Compositional inverse of Bessel function

Was ever studied a function $f$ which solves $J_0(f(x))=x$? Integral representations, natural domains of existence and whatever.

2
votes

0
answers

175
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### Integral representation of a quotient of odd and even parabolic cylinder functions

In some work on nonlinear splines in space, the following expression arises:
$$\frac{e^{-\mathrm{i}\pi/4} \; y_2 \left( \frac{\mathrm{i}}{4 \alpha} -\frac{1}{2} ; e^{\mathrm{i}\pi/4} \sqrt{\alpha} s \...

1
vote

1
answer

222
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### Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?

Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...

0
votes

0
answers

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### Incomplete Gamma function $\Gamma(0,x)$ and $\Gamma(0,-x)$

I want to find the value of this
\begin{align}
y=\Gamma(0,x)-\Gamma(0,-x)
\end{align}
where $\Gamma$ is the upper incomplete Gamma function, $x>0$ is real. I can't find the definition of $\Gamma(0,-...

0
votes

0
answers

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### Solution to the integral of Bessel $\int^1_0 x \sin(a x) J_1 (b x) dx$

I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral:
\begin{equation}
\int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...

3
votes

1
answer

155
views

### Is there a theory of "elementary closed form solution" at the operator level for differential equations?

We begin by considering the usual general first order linear equation of the form
$$ a_0 y' + a_1 y + a_2 = 0 $$
Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...

2
votes

3
answers

243
views

### Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$

I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$.
It seems there is no simple way even if we consider Lambert ...

2
votes

0
answers

92
views

### Existence of analytic function in disk algebra [closed]

Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?

1
vote

0
answers

149
views

### Special function: Pulse peak modified with a power term

PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...

1
vote

1
answer

73
views

### Solution to non-autonomous delay differential equation?

If you define a special function called the Lambert W function, you can explicitly solve the classic delay differential equation $x'(t) = x(t - a)$ by supposing the solution is some $\exp(\lambda t)$ ...

1
vote

0
answers

56
views

### How to extend this sum involving generalized harmonic numbers?

It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...

1
vote

2
answers

88
views

### Asymptotics of Bessel functions in the discrete parameter

Is there any information on the asymptotics of $J_n(z)$ as $n\to \pm\infty$ for fixed $z$ (real or imaginary)? I originally wanted to ask about the modified Bessel functions $I_n(z)$, but found out ...

2
votes

0
answers

88
views

### Evaluation of a summation involving Hermite polynomials

I started with the the following four-variable function, $f(s,x,y,u)$, expressed as a summation involving the product of Hermite polynomials.
$f(s,x,y,u)=\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{(n-...

6
votes

0
answers

170
views

### Example of a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$

I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ By non-trivially ...

0
votes

0
answers

83
views

### Solving a nonlinear equation maybe with Lambert W function

Can you please help me solve the following nonlinear equation?
\begin{equation}
\boldsymbol{z} \odot\left(\boldsymbol{\Gamma}^{\top} \boldsymbol{y}\right)=(\beta)^{\frac{1}{m-1}}\left(\frac{m-1}{...

0
votes

0
answers

68
views

### Hypergeometric function and a related inequality

Assume that $a>1$ and $k$ is a positive integer. How to prove that
$$\frac{k F[1,1+a,1+a+k,\frac{a}{1+a}]}{a+k}<1,$$ where $F$ is the Gauss hypergeometric function?

1
vote

0
answers

78
views

### Where this function attains its maximum?

I have been trying to prove that the function
\begin{equation}
(2-x)^{2k-p}\int_0^1t^{\frac{k}{p}-1}(1-tx)^{-\frac{k}{p}}dt, \,\,\, x\in [0,1]
\end{equation}
attains its maximum at $x=1$ under the ...

0
votes

0
answers

114
views

### Do polylogarithmic integral functions appear in the literature?

The logarithmic integral function is defined as follows: $$\operatorname{li}(x) := \begin{cases}
\int_{0}^{x} \frac{1}{\ln(t)} dt & \text{for } 0<x< 1 \\
PV \int_{0}^{x} \frac{...

3
votes

0
answers

144
views

### An inequality for integrals involving Laguerre polynomials

Let $k\ge n$ and $$A(k,n)=\frac{ \Gamma[1+k]}{n!\Gamma[1+k-n]^2}\int_0^\infty \frac{e^{-r}r^{k-n}}{L_n(-r)} dr$$
where $$L_n(-r) = \sum_{m=0}^n \frac{\Gamma(1+n)}{\Gamma(1+m)^2 \Gamma(1+n-m)}r^m$$ is ...