Questions tagged [special-functions]

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

4
votes
1answer
82 views

What is the Puiseux series of the Bessel function $J_n(n)$?

By numerical experimentation I find the first three terms of the Puiseux series of the Bessel function of the first kind $$ J_n(n) = \frac{\Gamma(\frac13)}{2^{2/3}\cdot 3^{1/6} \cdot \pi}n^{-1/3} - \...
4
votes
2answers
140 views

Hypergeometric series for $\mathrm{Cl}_2(\pi/3)$

I am trying to find a hypergeometric series for $\mathrm{Cl}_2(\pi/3)$, where $$\mathrm{Cl}_2(x)=-\int_0^x\log\left|2\sin\frac{t}2\right|dt=\sum_{k\geq1}\frac{\sin kx}{k^2}$$ Is the Clausen ...
10
votes
1answer
253 views

$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\frac{m_1 z}{a} + \frac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz.$

Question: How to find the closed-form solution for the given integral? $$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\dfrac{m_1 z}{a} + \dfrac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz,$$ where $k, a, b,...
1
vote
0answers
20 views

Evaluating the asymptotics of parabolic cylinder equations arising from coupled ODE's

For context, I'm studying the paper Coulomb blockade in superconducting quantum point contacts by Averin from 1998. Specifically, I am trying to find how he obtains equation 11 from equation 10, which ...
2
votes
0answers
64 views

Solving $1=\Gamma\left(\frac{d}{2}\right)-\Gamma\left(\frac{d}{2}, 2t \right)$

Is it possible to solve the following equation that contain the incomplete Gamma function? $$1=\Gamma\left(\dfrac{d}{2}\right)-\Gamma\left(\dfrac{d}{2}, 2t \right)$$ I would like to find the value ...
2
votes
2answers
58 views

On the functions $\mathrm{Gi}_{s}^{p,q}(x)=\sum\limits_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$

I have stumbled across the functions $$\mathrm{Gi}_s^{p,q}(x)=\sum_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$$ And I would like to know where I can learn more about them. These functions are interesting ...
2
votes
0answers
54 views

Reference Request: Parabolic Cylinder Functions (decaying at $+ \infty$), Order and Type in the Parameter

Background Reciprocal Gamma function as an entire function The reciprocal Gamma function, \begin{equation} \begin{split} \frac{1}{\Gamma(z)} &= e^{\gamma z} \prod_{n = 1}^{\infty} \left( 1 + \...
0
votes
0answers
25 views

Conjugate of Bessel functions of purely imaginary order

I would like to find a relation between $J_{i\nu}(x)$ and $J^*_{i\nu}(x)$ where $J$ are Bessel functions of the first kind, $*$ denotes the conjugate, and $\nu,x\in \mathbb{R}$ so that the functions ...
0
votes
0answers
33 views

If $L_{n}(x)$ is Laguerre's polynom, prove extension $ e^\frac{x}{2} = 2\sum_{n=0}^{\infty} (-1)^n \frac{L_{n}(x)}{n!} $

If $L_{n}(x)= e^x \frac{d^n}{dx^n}(x^n e^{-x}) $ is Laguerre's polynom, prove extension $ e^\frac{x}{2} = 2\sum_{n=0}^{\infty} (-1)^n \frac{L_{n}(x)}{n!} $ I really don't know how resolve this. I ...
13
votes
0answers
532 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
13
votes
1answer
303 views

Yet another difficult logarithmic integral

This question is a follow-up to MSE#3142989. Two seemingly innocent hypergeometric series ($\phantom{}_3 F_2$) $$ \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2\frac{(-1)^n}{2n+1}\qquad \...
1
vote
0answers
33 views

Putting some polynomials in terms of orthogonal polynomials

I came across some polynomials which take the form $$F_n(x)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{2k}{k}\binom{n-k}{k}x^{n-2k}.$$ I noticed that these look pretty similar to the series for the ...
0
votes
1answer
26 views

Can a logarithm function with two variable be expressed as a Meijer-G function?

Let {x},{y}>0 and {a},{b}>0. can the function log(1+ax+by) be expressed as a Meijer-G function?
0
votes
0answers
34 views

Spherical expansion of an exponential function?

We know that a normal planewave can be Rayleigh expanded by spherical harmonics as$$e^{i\vec k·\vec r}=4π\sum_{l=0}^∞\sum_{m=-l}^li^lj_l(kr)Y_{lm}(\hat{\vec k})Y_{lm}^*(\hat{\vec r}).$$ Does any body ...
0
votes
1answer
42 views

Power series associated with airy function

When $x$ goes to $+\infty$, the quotient $$\frac{N(x)}{D(x)}=\frac{x+\frac{2}{4!}\,x^4+\frac{2.5}{7!}\,x^7+\frac{2.5.8}{10!}\,x^{10}+\ldots}{1+\frac{1}{3!}\,x^3+\frac{1.4}{6!}\,x^6+\frac{1.4.7}{9!}\,x^...
1
vote
1answer
42 views

How does one compute the Euler product for the Dirichlet Beta function?

In this post, the author derives the Euler product for Dirichlet Beta function, defined as $$\beta(s) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^s}$$ for $\Re(s)>0$ and obtains $$\beta(s) = \prod_p ...
0
votes
1answer
42 views

Solve the Bessel differential equation

Show that $J_{n}(x) / x^{n}$ is a solution of $$\frac{d^{2} y}{d x^{2}}+\left(\frac{1+2 n}{x}\right) \frac{d y}{d x}+y=0$$ and that $\sqrt{(x)} J_{n}(k x)$ is a solution of $$\frac{d^{2} y}{d x^{2}}+\...
0
votes
1answer
24 views

checking the Solution of Bessel differential equation

I want to check the first part of the solution and help in the second part. Obtain the solution $$y_{1}(x)=J_{0}(x)=1-\frac{x^{2}}{2^{2}}+\frac{x^{4}}{2^{2}\cdot 4^{2}}-\ldots+\frac{(-1)^{n} x^{2 ...
2
votes
0answers
46 views

Jacobi polynomials and Gram determinants

On page 294, Andrews, Askey and Roy - Special functions. For sequences of (independent) functions $\lbrace \phi(x) \rbrace_{n=0}^{\infty}$ and $\lbrace \psi(x) \rbrace_{n=0}^{\infty}$, a sequence $\...
0
votes
2answers
15 views

Prove $B(p,q) + B(p+1,q) + B(p+2,q) + … = B(p,q-1) $ where $B$ is beta function and $q > 1$

Prove $B(p,q) + B(p+1,q) + B(p+2,q) + ... = B(p,q-1) $ where $B$ is beta function and $q > 1$. I tried with some basic formulas for beta function, mathematical induction but just cannot get some ...
13
votes
1answer
304 views

A logarithmic integral, generalization of a result of Shalev

As many of you are already aware, I and Marco Cantarini are currently working on the applications of fractional operators to hypergeometric series, extending the class of $\phantom{}_{p+1} F_p$s whose ...
1
vote
1answer
29 views

Do orthogonal polynomials determine the moments of their orthogonality measure?

I am currently learning about the inverse problem for orthogonal polynomials for orthogonality measures supported on the real line. My question is not about finding the orthogonality measure from the ...
1
vote
1answer
47 views

Evaluation of generalized Laguerre function integrals using orthogonality relations

(NB - I am not asking to be spoon-fed with complete solutions, just pointing out any useful transformations, or giving general pointers would suffice.) The orthogonality relation for generalized ...
0
votes
0answers
36 views

An equation with Gamma Euler function in critical strip

Let $$ D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \} $$ that is the critical strip without critical line. I have to find if the following equation, with ...
0
votes
0answers
30 views

Solution of a particular Trigonometric Integral

Does anyone have an idea how to solve this trigonometric integral? $$\int\frac{dx}{(a+bx)\sin(x)}$$ I have tried so many strategy but they don't work.
1
vote
0answers
48 views

Is $\int_0^1 \Psi(x)\Psi(1-x)\,dx$ related to any transform?

Is this related to any integral transform? $$\int_0^1 \Psi(x)\Psi(1-x)\,dx=\int_{0}^{1} e^{{\frac{1}{\log(x)}}+{\frac{1}{\log(1-x)}}} \, dx.$$ The integral, where $K$ is the modified Bessel function ...
0
votes
0answers
23 views

Error estimate in the approximation of Incomplete Beta Function

In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by $$f_{a,b}(x):=1-...
0
votes
0answers
19 views

Lattice associated to 4th Jacobi theta function?

For a lattice (specifically the dual lattice of a torus) there is associated a theta function $ \theta_{\Gamma}(w)=\sum_{\gamma\in\Gamma}w^{||\gamma||^2},\text{ where $w=e^{-4\pi^2t}$ and $t\in(0,\...
2
votes
0answers
50 views

Find the limit ‎ ‎$‎\displaystyle{\lim_{n\to\infty}}((x-m)‎\Gamma_q‎(n) + \sum_{k=1}^n ‎\Gamma_q‎(k) - \Gamma_q (k+x-m‎)) ‎$

The ‎q-‎gamma function ‎‎‎‎$‎‎\Gamma‎_q‎$‎ is defined as follows‎: ‎‎ ‎ ‎$‎‎\Gamma‎_q(x) =‎ ‎(1-q)^{1-x} ‎‎\prod‎_{n=0}^\infty ‎‎\frac{1-q^{n+1}}{1-q^{n+x}}‎‎‎$, ‎when ‎‎$‎|q|<1‎$‎. My question: ...
0
votes
0answers
34 views

Expanding a function in spherical coordinates

I have a function f(theta,phi,r) in spherical coordinates. The function dies out at r->infinity (r in my case is dimensionless). Is there a natural way of expanding the function, the same way a ...
1
vote
0answers
54 views

asymptotic decreasing function

I'm currently working on my university thesis, I'm trying to model a particular behavior but I can't think of the right function that works for me. I'm looking for: Decreasing function $f(0) = 1$ $...
0
votes
1answer
71 views

Expressing $G_{m,m+1}^{m+1,0}\left(x\middle| \begin{array}{c}1,\cdots,1 \\0,0,\cdots,0\\\end{array}\right)$ as a power series.

I have this family of MeijerG functions: $$ G_{m,m+1}^{m+1,0}\left(x\left| \begin{array}{c} 1,\cdots,1 \\ 0,0,\cdots,0 \\ \end{array} \right.\right) $$ which I'd like to express in terms of a power ...
9
votes
0answers
164 views

About the product of two Elliptic integrals

Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...
1
vote
0answers
45 views

Identity a Laplace Transform

I am looking for a function on the positive real line whose Laplace transform, with parameter $s$, is $$\left(\frac{\lambda}{1+\lambda}\right)^s,$$ where $s$ and $\lambda$ are greater than $0$. The ...
10
votes
1answer
210 views

A peculiar Euler sum

I would like a hand in the computation of the following Euler sum (Why isn't here a tag for Euler sums?) $$ S=\sum_{m,n\geq 0}\frac{(-1)^{m+n}}{(2m+1)(2n+1)^2(2m+2n+1)} \tag{1}$$ which arises from ...
2
votes
1answer
36 views

Special polynomials and an identity of hypergeometric series

Motivation: I have a few polynomials and am trying to find a representation for them in terms of special functions. I'm more interested in the techniques here, so I won't give any too particular ...
0
votes
0answers
27 views

Upper bound for the complex Beta function

Is there any work or reference regarding upper bounds for the complex beta function defined by \begin{equation} B(x,y)=\frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}, \end{equation} for $\Re{x} >0$ and $...
3
votes
1answer
50 views

A Continuous function $f: \overline{B_1(0)} \subset \ell^2\to \mathbb{R}$ which does not reach the maximum?

If necessary, recall that $$ \ell^2 = \{x=\{x_n\}_n\subset \mathbb{R} : \|x\|^2:=\sum_n |x|^2<\infty\} $$ and $ \overline{B_1(0)} $ is the closed unit ball with respect to that norm. Can we ...
4
votes
2answers
71 views

Calculate sum of series ‎$‎\sum_{k=1}^\infty (k+x-m)^\alpha e^{\beta (k+x-m)}$‎.

‎I've been stuck with calculating the sum of series of the following problem. Can you help me?‎ ‎ $‎\sum_{k=1}^\infty‎(k+x-m)^\alpha ‎e^{‎\beta‎(k+x-m)}‎$ for ‎$‎‎\alpha‎>0‎$‎, ‎$‎‎\beta‎<0‎$‎...
15
votes
2answers
420 views

“Continuized” Taylor Series? $\sin(x)=\sum \frac{(-1)^nx^{2n+1}}{(2n+1)!}=\int_{-1}^\infty \frac{\cos(\pi n) x^{2n+1}}{G(2n+1)}dn$?

~~not trying to reinvent the Laplace transform, but just an exploration into these particular series and integrals~~ Current answers don't fully address the 5 questions, so any new ideas or ...
0
votes
0answers
34 views

A conditional expectation of the beta binomial distribution?

Consider a beta binomial distribution where the number of trials, $n$, is odd and the shape parameters of the underlying beta distribution, $\alpha$ and $\beta$, are equal. Is there a closed form ...
1
vote
1answer
92 views

Real Analysis Methodologies to Prove $-\int_x^\infty \frac{\cos(t)}{t}\,dt=\gamma+\log(x)+\int_0^x \frac{\cos(t)-1}{t}\,dt$ for $x>0$

In this question, the OP asked to prove the Cosine Integral identity $-\int_x^\infty \frac{\cos(t)}{t}\,dt=\gamma+\log(x)+\int_0^x \frac{\cos(t)-1}{t}\,dt$, where $\gamma $ is the Euler-Mascheroni ...
2
votes
2answers
77 views

Why are there two different recurrences for Gegenbauer polynomials?

As I mentioned previously, I've been reading up on Gegenbauer polynomials in preparation for a blog post on the kissing number problem—specifically, the Delsarte method. To make a long story short, ...
3
votes
0answers
44 views

Can every Gaussian integral be reduced to elementary functions and poly-logarithms only?

Let us define a following function: \begin{eqnarray} {\mathcal J}^{(d)}(\vec{A}) := \int\limits_0^\infty e^{-u^2} \prod\limits_{\xi=1}^d erf(A_\xi u) du \end{eqnarray} for $\vec{A}:=\left(A_\xi\right)...
0
votes
0answers
71 views

Integral involving the logarithm of a confluent hypergeometric function

I am trying to find the solution of the integral \begin{align} I =\int_{0}^{\infty}e^{-t}t^{\alpha+1}\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}\log\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}dt \...
5
votes
0answers
104 views

Is a closed form possible for $\int\frac{\text{Li}_2(x)^2}{x}dx$?

Can $\,\displaystyle\int\frac{\text{Li}_2(x)^2}{x}dx\,$ be calculated by a sum/term of polylogarithm functions and the natural logarithm and polynomials (“closed form”) ? For the special case $\,\...
1
vote
0answers
23 views

Sum involving hypergeometric 2F2 function

I'm trying to simplify the following sum: $$ \sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!}, $$ where $n=1,2,\ldots$, $z>0$, $0<a<1$,...
1
vote
0answers
15 views

Integrate a Generalized incomplete gamma function

My question is about an integral but not an ordinary one $\int_{le^{-bt'}}^{l}s^{l-1}e^{-s}ds$ I have the idea of use the leibniz integral rule because I don't want to use the Generalized ...
0
votes
1answer
95 views

An integral involving a Gaussian, error functions and the Owen's T function.

This question is closely related to An integral involving a Gaussian and an Owen's T function. and An integral involving error functions and a Gaussian . Let $\nu_1 \ge 1$ and $\nu_2 \ge 1$ be ...
0
votes
1answer
28 views

Parabolic cylinder functions

My question is the same as this question but more general. I am dealing with parabolic cylinder functions and misunderstand some moments. As the source, I use this. The authors state that these ...