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Questions tagged [special-functions]

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

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29 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 ...
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1answer
14 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?
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25 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 ...
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1answer
34 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^...
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1answer
24 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 ...
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1answer
21 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}}+\...
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1answer
19 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 ...
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0answers
26 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 $\...
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2answers
13 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 ...
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45 views

Find the limit of ‎$‎\log‎\prod‎_{n=1}^\infty ‎\frac{n(n+x+a)}{(n+a)(n+x)}‎$. [closed]

‎‎Consider the following production $‎‎‎‎‎\log‎‎‎\prod‎_{n=1}^\infty ‎‎\frac{n(n+x+a)}{(n+a)(n+x)}‎‎‎$, for ‎$‎a‎‎>0‎$‎ and ‎$‎x>0‎$‎.‎‎‎‎‎‎‎ ‎‎‎ Can anyone find the limit of production?‎
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1answer
222 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 ...
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1answer
18 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 ...
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1answer
36 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 ...
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0answers
33 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 ...
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0answers
24 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.
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0answers
41 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 ...
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0answers
18 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-...
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11 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,\...
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0answers
42 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: ...
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0answers
26 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 ...
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0answers
41 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 $f(0) = 1$ $...
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1answer
62 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 ...
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0answers
140 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}...
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0answers
31 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 ...
9
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1answer
185 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 ...
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1answer
30 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 ...
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0answers
16 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 $...
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1answer
46 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 ...
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2answers
68 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 ‎$‎‎\alpha‎>0‎$‎, ‎$‎‎\beta‎<0‎$‎...
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2answers
384 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 ...
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0answers
17 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
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1answer
80 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 that $-\int_x^\infty \frac{\cos(t)}{t}\,dt=\gamma+\log(x)+\int_0^x \frac{\cos(t)-1}{t}\,dt$, where $\gamma $ is the Euler-Mascheroni constant. However, the two ...
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0answers
36 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)...
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0answers
62 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 \...
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15 views

function that generates a single peak in a graph

as stated i'm simply trying to generate a singular peak in a graph, my first solution works well enough but is rather long, but i'm wondering if there's any shorter function that could do just that $$...
4
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0answers
86 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 $\,\...
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0answers
16 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 $n=1,2,\ldots$, $z>0$, $0<a<1$,...
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0answers
13 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 ...
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1answer
52 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 ...
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1answer
24 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 ...
3
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1answer
73 views

Special Functions defined by Integrals.

There was this integral which caught my attention, when I was checking out The Applications of Beta and Gamma Functions. So, how can i prove the below, using change of variable? $$\int_{0}^{1}\frac{...
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3answers
159 views

Asymptotic expansion of $\int_0^1 \frac{\operatorname{K}(r x)}{\sqrt{(1-r^2 x^2)(1-x^2)}} \, \mathrm{d} x $

Notation: For $\varphi \in [0,\frac{\pi}{2}]$ and $k \in [0,1)$ the definitions $$ \operatorname{F}(\varphi,k) = \int \limits_0^\varphi \frac{\mathrm{d} \theta}{\sqrt{1-k^2 \sin^2(\theta)}} = \int \...
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0answers
14 views

Completeness relation for Jacobi Polynomials

I was wondering if there exists a completeness relation for Jacobi Polynomials, $P^{\alpha,\beta}_{n}(x)$ as in the case of Hermite polynomials, $H_{n}(x)$ such that $$ \sum^{\infty}_{n=0} \psi_n(x) \...
1
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1answer
52 views

How to determine $\gamma$ in Fox H-function

In the following Fox H-function the contour $L$ is either $L_{-\infty}$, $L_{+\infty}$ or $L_{i\gamma\infty}$. $$ H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) &...
5
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1answer
85 views

Polylogarithm inequality: $(s+1)\frac{-\operatorname{Li}_{s+1} (-x)}{-\operatorname{Li}_s(-x)} > \log(x)$

For $s \geq 0$ and $x > 0$ define $$ f_s (x) = - \operatorname{Li}_s (-x) \stackrel{s > 0}{=} \frac{1}{\Gamma(s)} \int \limits_0^\infty \frac{x t^{s-1}}{\mathrm{e}^t + x} \, \mathrm{d} t \, .$$ ...
0
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0answers
28 views

What's the generalised formula for this sequence?

Suppose we have two integers 'n' and 'k' and we have to find the formula for the sequence: ...
0
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2answers
100 views

Imaginary part of $\frac{\Gamma'(z)}{\Gamma(z)}$ [closed]

I want to prove that: $$ \Im\left(\frac{\Gamma'(z)}{\Gamma(z)}\right) = 0 \qquad \textit{iff} \qquad \Im(z) = 0$$ I tried to plot the function on mathematica and it seems true, excluding the poles ...
1
vote
1answer
40 views

Infinite Integral of a Product of Bessel Functions

I am interested in any analytic information about the following integral: $i^{4m+1} \int_0^{\infty} t^{1/4} J_m^4(t) J_{\nu}(\alpha t) dt$ where $i = \sqrt{-1}$ is the imaginary unit $m$ is a ...
7
votes
1answer
123 views

closed form for $\int_0^1\frac{\mathrm{Li}_s(x-x^2)}{x-x^2}\mathrm dx$

I am trying to evaluate $$F(s)=\sum_{n\geq1}\frac1{n^{s+1}{2n\choose n}}$$ I started off by noting that $$\frac1{n{2n\choose n}}=\frac12\int_0^1\left[x-x^2\right]^{n-1}\mathrm dx$$ So $$F(s)=\int_0^1\...
1
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1answer
52 views

How to express cosine of Fourier series as Fourier series again

I have the following Fourier series exapansion: \begin{equation} \phi(t) = a_0 + \Sigma_{n=1}^\infty (a_n\cos pnt + b_n\sin pnt). \end{equation} I want to express $\cos(\phi(t))$ as Fourier series ...