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.

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2
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0answers
48 views

Separating a Riemann-Hilbert problem

Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is $$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$ where $g_j(x)$ are Holder continuous functions and $...
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2answers
116 views

A recurrence formula for the Legendre function $P_\mu^\nu(x)$

Im looking for a recurrence formula of type: $$(\mu-\nu) x P_\mu^\nu(x) + P_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$ where $P_\mu^\nu(x)$ is the Legendre function of the first kind (solution to ...
11
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2answers
465 views

Do infinitely nested radicals have any applications?

There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
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34 views

Prove the supremum of Mittag-Leffer function

I find two interesting limits : \begin{align*} \frac{1}{2}& =\lim_{s\to 1^-}\sum_{n=0}^{\infty}\left(-1\right)^n\frac{\Gamma(1+ns)}{\Gamma(1+n)}\\ & =\lim_{s\to 1^+}\sum_{n=0}^{\infty}\left(-1\...
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50 views

Limit of Hankel function for large complex order, fixed real argument

Consider the Hankel function $H_\nu(z)$ where $\nu=re^{i\theta}$ (real $z>0$, $r>0$, $0\leq\theta<\pi$) as $r\rightarrow\infty$. I am aware that the Bessel function $J_\nu(z)$ has the ...
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45 views

Question about generalized Lambert's function as solution of a power tower with nested radical

I ask here because I want to draw attention on a question https://math.stackexchange.com/questions/3607819/special-power-tower-x-sqrtxx-sqrtxx-sqrt-cdots-and-generalize . I'm interested by the ...
7
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1answer
301 views

A variant of Lambert function

How to express the solution of $x^{x+1}=a$ using Lambert function? I know that the standard Lambert function can be used to describe the solution of $x^x=a$. I wonder if $x^{x+1}=a$ can be addressed ...
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On various versions of the harmonic oscillator

The standard $n$-dimensional harmonic oscillator is the operator $ \mathcal H=\frac{1}{2}\sum_{1\le j\le n}(D_j^2+x_j^2), \text{ $D_j=-i\partial_{x_j}$}, $ and its spectral decomposition is $$ \...
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75 views

Closed form for a double integral over the first quadrant of the $L^p$ disk

Is there, by any chance, a closed form for the following integral $$ I_p=\iint_{Q_p}(x+y)\log(x+y)dxdy, $$ where $Q_p=\{(x,y)\in\mathbb{R}^2, x>0,y>0,x^p+y^p\leq1\}$, $0<p\leq\infty$ ? ...
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Constructing a Meromorphic function on a genus 1 surface with prescriped divisor using Jacobi elliptic functions

I need to construct a meromorphic function $f \in \mathcal{M}(X)$ using the Jacobi elliptic functions on a genus $1$ surface with divisor of the form: $(f) = n \cdot P_0 + P_1 - n \cdot Q_0 - Q_1$, ...
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186 views

Degree of automorphic forms, SL(3,Z), and the elliptic Gamma function

In this article, the authors interpret a certain special function, the elliptic Gamma function, defined as $$ \Gamma(z,\tau,\sigma)=\prod_{j,k=0}^\infty\frac{1-e^{2\pi i((j+1)\tau+(k+1)\sigma-z)}}{1-...
1
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1answer
527 views

If $x^x=2$ then is $x$ expressible using elementary functions?

I have a curious question. Let $x∈\mathbb{R}^+$ such that $x^x=2$. I am aware that the Gelfond–Schneider theorem implies that $x$ cannot be algebraic. However, is it still possible that $x$ can be ...
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52 views

Construct a doubly periodic function on $\mathbb{C}$ using Jacobi elliptic functions with anti-holomorphic involution

I would like to find explicit examples of non-constant meromorphic doubly periodic $f(z)$ on $\mathbb{C}$, i.e. a meromorphic function on $\mathbb{C} / \Gamma$ for some lattice $\Gamma$, such that ...
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33 views

Bounding the working precision required in Spouge's Approximation

Spouge's approximation for the gamma function is $\Gamma(z+1) = (z+a)^{z+\frac{1}{2}}e^{-z-a} \left(c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \epsilon_a(z) \right)$ where the coefficients are given ...
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1answer
84 views

Integral expressions for Bessel-like power series

I'm interested in power series of form $$f(z)=\sum_{k=0}^\infty \frac{z^k}{(k!)^\alpha}.$$ When $\alpha=1$, this becomes $\exp(z)$. For $\alpha=2$ this is a Bessel function and for larger integer $\...
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2answers
90 views

First-order non-linear differential equation and transcendental equation

I'm trying to solve this differential equation : $$ \frac{dy}{dx}= \frac{-2 y^3}{(y+1)^2(y+2)^2} $$ with the boundary condition $y(x_0)=x_0$, $x>0$, and $y(x)$ being a positive function. The ...
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2answers
92 views

The exact constant in the simple bound of the fraction of Gamma Functions

In the Question : Upper bound of the fraction of gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
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Trying to bound the generalized hypergeometric function ${}_2F_3(x+1,x+1;1,1,1;\alpha)$ as $x\to \infty$?

(See also edit below)... I am trying to get a nice, explicit, bound on the hypergeometric function $$ {}_2F_3(a_1,a_2;b_1,b_2,b_3;\alpha), $$ in the case of a large parameter. In particular I am ...
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62 views

irrationality of Bessel function in $p$-adic

Let $J_0(z)=\sum_{n\ge 0}\frac{(-1)^n}{n!^2}\left(\frac z2\right)^{2n}$ be the Bessel function considered in $\mathbb C_p$. Let $\alpha\in\mathbb Q^*$ be in the convegence disk of $J_0$. Is $J_0(\...
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1answer
111 views

Integral involving associated Laguerre polynomial and Bessel function

In a quantum mechanics problem I encountered the following integral \begin{equation*} \int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,, \end{equation*} where $L$ denotes the ...
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1answer
130 views

Can we characterize the set of neoclassical production functions?

INTRODUCTION The neoclassical production function is the main building block in neoclassical growth theory, and consequently the main building block of modern macroeconomic theory. Mathematically, ...
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Conceptual meaning of a non-linear relation connecting $6$ Mordell integrals?

Define Mordell integral by $$ \phi_\alpha(\theta)=\int\limits_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx.\tag{1} $$ There are a lot of linear relations connecting integrals ...
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1answer
127 views

Polylogarithm : reference request for proof of integral representation

On page 494 of the book Integrals and series, volume I : elementary functions, Gordon and Breach, 1986, by A. P. Prudnikov, Y. A. Brychkov, and O.I. Marichev, (perhaps translated from Russian), the ...
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36 views

Explanation of the asymptotic expansion of $Ai(x^2)$ by steepest

I am reading Copson's textbook Asymptotic Expansions, on page 100 he writes the following passage: We then get: $$(39.1) Ai(\nu^2)=\frac{\nu}{2\pi i} \exp(2\nu^3/3)\int_{-\infty}^{\infty}\exp(\nu^3 ...
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183 views

On an inequality involving the Lambert $W$ function and the sum of divisors function

Let $W(n)$ be the principal/main branch of the Lambert $W$ function (this is the Wikipedia related to this special function). I was inspired in Robin equivalence to the Riemann hypothesis (see [1]) ...
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1answer
256 views

Reciprocal expansion of modified Bessel function

I am reading Sherstyukov and Sumin - Reciprocal expansion of modified Bessel function in simple fractions and obtaining general summation relationships containing its zeros. The authors say they are ...
4
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131 views

Expressing the inverse Dixon function in terms of more familiar functions

If $x^3+y^3-3\alpha xy=1$, is there an expression for the integral $$\int_0^z \frac{\mathrm dx}{y^2-\alpha x}$$ in terms of more familiar functions? A.C. Dixon introduced the elliptic functions $\...
4
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1answer
199 views

Gegenbauer's addition theorem for Jacobi polynomials

I have the following identity, $$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z) = 2 \, j_n(x) \, j_n(y) \;,$$ where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...
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1answer
196 views

Reduction of Meijer G-function

I would like to simplify the following Meijer G-function: $ G_{1,2}^{1,1} \left(z\mid \binom{0}{0,a}\right) $ into a new Meijer G-function of lower order. In other words, I would like to "simplify" ...
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1answer
62 views

The ratio of Hankel functions

I have to obtain an asymptotic solution for small real positive $x$ for the ratio of Spherical Hankel functions ($n=0,1,2....)$ ${h^{(2)}_n(x)}/{h^{(1)}_n(x)}$ I found that series should be $-1 + i ...
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1answer
140 views

The existence of an interval $I\subset (0.8856,+\infty)$ such that the derivative $(\Gamma^{-1})’(x)$ is greater than $1$ [closed]

The principal inverse function of the gamma function is denoted by $\Gamma^{-1}$. See the paper: Uchiyama - The principal inverse of the gamma function. $\Gamma^{-1}$ is an increasing and concave ...
2
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2answers
125 views

Simplify the difference of two dilogarithms--as in the logarithmic counterpart

This question--pertaining to the quantum-information-theoretic topic of "bound entanglement"--stems from the question and answer to https://math.stackexchange.com/questions/3464105/if-possible-...
2
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1answer
40 views

Reference request: sums of rational functions and polygamma functions

I have heard that there are ways to express sums of rational functions in terms of polygamma functions, and I would like to read more about it. However, I don't know the literature about special ...
25
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2answers
2k views

A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula

On the Wolfram page about the Euler-Mascheroni Constant $\gamma $, the following amazing limit is given without proof (referring to "personal communication"): $$\lim_{z\to\infty}\left[\zeta(\zeta(z))-...
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1answer
150 views

Logarithm of an integral involving generalized real binomial coefficients

I could not find a closed form for this integral although I think it should have been studied. What is a good approximation to $I$ in $$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...
0
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1answer
72 views

Iterating the the ODE for Bessel function

If we look at the Bessel ODE: $$x^2 y'' + xy' + (x^2 - \alpha^2)y = 0$$ Suppose I then put the solution to the above ODE as $J_{\alpha}(x)$ in the RHS, and try to solve the following ODE: $$x^2 y'' + ...
2
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1answer
79 views

Can the integral $ \int_0^R\quad J_{m-n}(a r)J_m(b r) dr$ be explicitly represented in a closed form?

Doe the following definite integral have an explicit representation in terms of a Bessel functions or a generalized hypergeometric function ${}_pF_q$? $$ \int_0^R\quad J_{m-n}(a r)J_m(b r) dr, \quad \...
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0answers
31 views

Fourier Transformation of Generalized Laguerre Functions

Where can I find, or How to calculate the discrete fourier transformation of the following special functions? $$\begin{equation}\begin{aligned} &\frac{1}{\sqrt{N}}\sum_{\vec{R}_{j}}e^{i\vec{k}\...
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0answers
71 views

Asymptotic approximations or upper bounds for ${}_{2}F_{1}(x+1,x+1,1,z)$ when $x \gg 1$?

I have recently encountered the hypergeometric function $$ {}_{2}F_{1}(x+1,x+1,1,z), $$ where $x$ is an integer and $z$ is a real number with $x \ge 1$ and $0<z<1/2$. This is the first time I ...
5
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1answer
310 views

Convergence of the series of Legendre polynomials

Consider the generating function of Legendre polynomials: $$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum\limits^{\infty}_{n=0} P_n(x)t^n$$ Is it true that for $0<x<1, t=1$ series of Legendre ...
1
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1answer
129 views

Estimation of Hypergeometric function ${_3F_2}$ [closed]

Is there any way to estimate the following function, which is a result of sum of ratios of Gamma functions? $$ {_3F_2}\begingroup \renewcommand*{\arraystretch} % your pmatrix expression \left[ \begin{...
3
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1answer
162 views

Real and imaginary parts of $\ln \Gamma(i b)$

The imaginary part of the digamma function when its argument is pure imaginary is known as $$\Im\psi(\mathrm{i}b)=\frac{1}{2}b^{-1}+\frac{1}{2}\pi\coth{\pi b},$$ and its real part is much more ...
2
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0answers
132 views

Monotone coupling between “two-sided Gumbel” distributions

I am interested in finding a monotone coupling between two random variables. Let $\alpha_1>\alpha_2$, $b<a$. Define the following two (non-normalized) densities on the whole real line: \begin{...
1
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1answer
79 views

Original examples of functions of slow increase in the spirit of Jakimczuk

I believe that it is possible to prove that $$f(x)=e^{\operatorname{Ai}(x)}\log x$$ is a function of slow increase in the spirit of the definition given by the author of [1], where $\operatorname{Ai}(...
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0answers
32 views

Reflection formula for the ${}_2\!F_1$ hypergeometric function of a matrix argument

According to my implementation of the hypergeometric function of a matrix argument, the so-called "Reflection formula" for ${}_2\!F_1$ given on DMLF (formula 35.7.8) is not true. On the Wikipedia ...
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46 views

Why do we define the hypergeometric function of a matrix argument for symmetric matrices only?

The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X)...
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0answers
18 views

Does the hypergeometric function of a matrix argument depend on $\alpha$ for a $1\times 1$ matrix?

I already posted this question on maths.SE but got no answer. The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\...
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0answers
40 views

Upper bound to the Wright omega function

I am looking for a reference describing upper bounds to the Wright omega function. Thanks very much in advance !
43
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1answer
932 views

Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?

Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs. I encountered other analogies as well. $${\begin{aligned}\gamma &=\int _{0}^{1}\int _{...
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0answers
74 views

Macdonald's idea of his kth weight

This question is about Macdonald's symmetric polynomials theory. Going through related papers and literature, it seems to me that the magical part of his theory lies in how the k-th weight function $\...

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