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
2
votes
0
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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}\...
- 11.3k
5
votes
1
answer
205
views
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-...
- 11.3k
-2
votes
1
answer
135
views
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(...
- 3
2
votes
2
answers
174
views
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 ...
- 11.3k
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^...
- 11.3k
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 + ...
- 11.3k
-1
votes
0
answers
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) + ...
-1
votes
0
answers
33
views
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?
- 1
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 ...
- 12k
1
vote
0
answers
119
views
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}\...
- 706
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,
$$
where
$\phi_1(\alpha)=\frac{\alpha}{2}
\left\{{\{\Gamma(3/\alpha)\}^{1/...
- 148
4
votes
4
answers
494
views
What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?
It is known that
\begin{equation*}
\tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation*}
and
\begin{equation*}
\ln\tan x=\ln x+\...
- 706
2
votes
0
answers
95
views
Asymptotic expansion of Jacobi function
For $\alpha,\beta \in \mathbb{C},\, \alpha$ a non-negative integer, we define $$A_{\alpha,\beta}(t)=(\sinh t)^{2\alpha+1}(\cosh t)^{2\beta+1} $$
and $$ \mathcal{L}_{\alpha,\beta}=\frac{d^2}{dt^2}+\...
- 123
4
votes
0
answers
120
views
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{...
- 789
1
vote
0
answers
96
views
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 ...
- 19
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 ...
- 1
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 ...
- 317
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 $B_n(x)$. ...
- 8,838
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
$$I= \int_0^\infty \sqrt x e^{\large -4/3x^{3/2}}\left(\int_0^x \operatorname{Ai}(t)dt\int_0^x \operatorname{Bi}(t)dt\right)dx$$
$$=\frac{1}{3}-\frac{\sqrt[3]{2\sqrt 3+3}+\sqrt[3]{2\sqrt 3-3}...
- 175
5
votes
2
answers
287
views
Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$?
By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$):
$$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 \...
- 1,125
0
votes
1
answer
193
views
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 $n \geq 2$,
$$
\sum_{m=1}^n S(n, m) (-1)^m (m-1)!...
- 58
2
votes
0
answers
96
views
Power series of the modified Bessel function of the second kind
I am looking for a power series representation of
$$ \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 ...
- 83
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 ...
- 3
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 ...
- 1
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 ...
- 595
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 ...
- 11
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 ...
- 706
2
votes
1
answer
122
views
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) ...
- 706
4
votes
1
answer
110
views
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.
- 94.7k
2
votes
0
answers
175
views
Integral representation of a quotient of odd and even parabolic cylinder functions
In some work on nonlinear splines in space, the following expression arises:
$$\frac{e^{-\mathrm{i}\pi/4} \; y_2 \left( \frac{\mathrm{i}}{4 \alpha} -\frac{1}{2} ; e^{\mathrm{i}\pi/4} \sqrt{\alpha} s \...
1
vote
1
answer
222
views
Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?
Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
- 379
0
votes
0
answers
48
views
Incomplete Gamma function $\Gamma(0,x)$ and $\Gamma(0,-x)$
I want to find the value of this
\begin{align}
y=\Gamma(0,x)-\Gamma(0,-x)
\end{align}
where $\Gamma$ is the upper incomplete Gamma function, $x>0$ is real. I can't find the definition of $\Gamma(0,-...
- 11
0
votes
0
answers
80
views
Solution to the integral of Bessel $\int^1_0 x \sin(a x) J_1 (b x) dx$
I've been trying to work out the solution of this integral. I have seen in the Gradshteyn (6.669(9)) a similar integral:
\begin{equation}
\int_0^1 x^\nu \sin(a x)J_\nu(a x)dx= \frac{1}{2\nu+1}\left[\...
3
votes
1
answer
155
views
Is there a theory of "elementary closed form solution" at the operator level for differential equations?
We begin by considering the usual general first order linear equation of the form
$$ a_0 y' + a_1 y + a_2 = 0 $$
Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
- 1,450
2
votes
3
answers
243
views
Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$
I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$.
It seems there is no simple way even if we consider Lambert ...
2
votes
0
answers
92
views
Existence of analytic function in disk algebra [closed]
Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
- 149
1
vote
0
answers
149
views
Special function: Pulse peak modified with a power term
PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
- 741
1
vote
1
answer
73
views
Solution to non-autonomous delay differential equation?
If you define a special function called the Lambert W function, you can explicitly solve the classic delay differential equation $x'(t) = x(t - a)$ by supposing the solution is some $\exp(\lambda t)$ ...
- 217
1
vote
0
answers
56
views
How to extend this sum involving generalized harmonic numbers?
It is well-known since Euler that the Generalized harmonic numbers, defined for $n\in\mathbb N$ by $$H_n^{(r)}=\sum_{k=1}^n\frac1{k^r},$$ can be naturally extended for non integer $n$ in terms of ...
- 12.8k
1
vote
2
answers
88
views
Asymptotics of Bessel functions in the discrete parameter
Is there any information on the asymptotics of $J_n(z)$ as $n\to \pm\infty$ for fixed $z$ (real or imaginary)? I originally wanted to ask about the modified Bessel functions $I_n(z)$, but found out ...
- 1,831
2
votes
0
answers
88
views
Evaluation of a summation involving Hermite polynomials
I started with the the following four-variable function, $f(s,x,y,u)$, expressed as a summation involving the product of Hermite polynomials.
$f(s,x,y,u)=\sum_{n=0}^{\infty}\frac{H_{n}(x)H_{n}(y)}{(n-...
- 21
6
votes
0
answers
170
views
Example of a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ By non-trivially ...
- 421
0
votes
0
answers
83
views
Solving a nonlinear equation maybe with Lambert W function
Can you please help me solve the following nonlinear equation?
\begin{equation}
\boldsymbol{z} \odot\left(\boldsymbol{\Gamma}^{\top} \boldsymbol{y}\right)=(\beta)^{\frac{1}{m-1}}\left(\frac{m-1}{...
0
votes
0
answers
68
views
Hypergeometric function and a related inequality
Assume that $a>1$ and $k$ is a positive integer. How to prove that
$$\frac{k F[1,1+a,1+a+k,\frac{a}{1+a}]}{a+k}<1,$$ where $F$ is the Gauss hypergeometric function?
- 31
1
vote
0
answers
78
views
Where this function attains its maximum?
I have been trying to prove that the function
\begin{equation}
(2-x)^{2k-p}\int_0^1t^{\frac{k}{p}-1}(1-tx)^{-\frac{k}{p}}dt, \,\,\, x\in [0,1]
\end{equation}
attains its maximum at $x=1$ under the ...
0
votes
0
answers
114
views
Do polylogarithmic integral functions appear in the literature?
The logarithmic integral function is defined as follows: $$\operatorname{li}(x) := \begin{cases}
\int_{0}^{x} \frac{1}{\ln(t)} dt & \text{for } 0<x< 1 \\
PV \int_{0}^{x} \frac{...
- 3,939
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 ...
- 115