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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158 views

What conditions imply that a function over $\mathbb{Z}$ is a polynomial?

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...
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0answers
118 views

Conway's box function iterated to produce a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
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0answers
48 views

Two integrals involving Legendre Functions

I have two Integrals which I want to identify with a simpler functions (if possible). Firstly, $$\int_{1}^{\infty} \frac{1}{\sqrt{x^2-1}}Q_{-1/2+k}\left(\frac{x\cdot a-b}{\sqrt{x^2-1}} \right)\cdot ...
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0answers
61 views

Energy Oscillations in a One Dimensional Crystal

Good day! Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially ...
0
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0answers
100 views

Four kinds of generalized hypergeometric formulas for $\pi$

Given, $$\begin{array}{|c|c|c|c|} \hline n&a_n&b_n&c_n\\ \hline 1 & 6541681608 & 163096908 & -640320^3\\ \hline 2 & 85840 & 4492 & -14112^2\\ \hline 3 & 28302 ...
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262 views

What is a “generalized zeta function”?

Out of procrastination I computed $$\sum_{k=1}^\infty k^{-k^2}\sim 1.06255080549625593786944593879.$$ The inverse symbolic calculator identified this number as "From generalized Zeta function". I do ...
5
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2answers
288 views

A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where $$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$ is the Lerch ...
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0answers
50 views

how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written: $$ \frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1} $$ and so by symmetry: $$ \frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} ...
-2
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2answers
147 views

On Bohr-MollerupTheorem [closed]

In http://mathworld.wolfram.com/Bohr-MollerupTheorem.html, Bohr-Mollerup Theorem is given where it is stated that $\Gamma$ function is the unique log convex function that satisfies ...
4
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68 views

An identity of complicated combinations of gamma functions (related to hypergeometric functions)

Can somebody help me in proving the following equation? \begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) ...
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162 views

Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ ...
0
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1answer
74 views

Upper bound for a ratio of modified Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...
5
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0answers
120 views

Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by $$ I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta $$ and arises, among other places, in the probability density function of a ...
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0answers
189 views

What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like: For $k, l \in \mathbb{Z}$, we have $k! \times ...
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0answers
90 views

What is $\int (1-e^{-x})^n dx$? [closed]

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
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1answer
149 views

How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...
1
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2answers
134 views

Is there any simpler form of this function

Assume that $n$ is a positive integer. Is there any simple form of this hypergeometric value $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$
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0answers
105 views

A hypergeometric identity [closed]

Is there any simple proof of this identity $${_4F_3}[\{\frac{1}{2}+\frac{n}{4},1+\frac{n}{4},-p,p\},\{\frac{1}{2},\frac{3}{2},\frac{1}{2}+\frac{n}{2}\},1]=\frac{\Gamma[\frac{1+n}{2}] ...
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1answer
106 views

Approximation of $ _2F_1((b-1)a,b;ba;x) $

Is there any (simple) approximation of this Hypergeometric function: $ _2F_1((b-1)a,b;ba;x) $, where $0<x<1$ and $b>a>1$. Thanks!
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0answers
70 views

q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$ are orthogonal in $\theta \in [0, \pi]$ with ...
2
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2answers
142 views

How to calculate one Cauchy type determinant

As we know, a Cauchy determinant of size n admits the following explicit formula: $$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y ...
4
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1answer
520 views

How to prove this identity on double summation series?

I suspect the following identity is valid, but I can not prove it. I just calculate it numerically. ...
0
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1answer
102 views

Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared: $$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$ where $\rho,s>1$, ...
5
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201 views

Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials: $$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$ Usual arguments from the theory of formal groups allow to prove that for any $n$ ...
3
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0answers
154 views

Analysing functions on zero-length intervals and super-small values

Suppose a function that has a pole in $x=0$: Here we see the graphic of the real part of the Gamma function. As we can see on it, there is a vertical line at $x=0$ that comes from $-\infty$ to ...
5
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121 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
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0answers
110 views

Integral involving a Meijer-G function

I am having trouble with calculating the following integral: $$ \int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx, $$ where $\alpha > ...
2
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1answer
143 views

Eigenfunctions of an infinite summation operator

I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*: $f \rightarrow \sum_{1}^{\infty} f(nx)$ So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$ ...
3
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0answers
141 views

System of linear ODEs with hypergeometric coefficients

For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$: $$ \begin{align}x ...
2
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2answers
236 views

Is this infinite series related to some well-known special functions?

Please allow me to resort once again to the expertise of the MathOverflow community : During research I encoutered the following infinite series : $$\sum_{n=-\infty}^{+\infty} ...
4
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1answer
204 views

Asymptotic behaviour of $K$-Bessel function in transition range

It is known that the famous mistake of Iwaniec-Sarnak in their paper of $L^\infty$ norm of eigenfucntion of non-cocompact arithmetic surfaces in lemma (A1) is because of they did not consider the bump ...
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38 views

Request for reference about bound on zeroes of the Laguerre polynomials

Consider the sequence of polynomials given as, $p^{a}_k (x) = (1 - a \frac{d}{dx})^k x^n $ for some parameter $a>0$ and $k$ being a positive integer. For any positive integer $d$ it seems to be ...
4
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2answers
255 views

What is known about this series?

I recently came across the following function which intrigues me: \begin{equation} f(\alpha):=\sum_{i=0}^\infty \frac{\alpha^{i(i+1)/2}}{i!}. \end{equation} For $-1\leq \alpha\leq 1$ this function is ...
6
votes
2answers
316 views

expression for infinite series with powers of factorial in denominator

The series $$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$ has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...
4
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1answer
92 views

Estimate on sum of $J_n^4$

If $J_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum_{n=-\infty}^{\infty} J_n^2(x)=J_0^2(x)+2\sum_{n=1}^{\infty} J_n^2(x)=1.$$ What is known about $$ ...
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1answer
78 views

Maximal minimum of Bessel functions

This comes from a scattering problem. Consider the usual non singular Bessel functions of the first kind, $J_n(x)$. It is known that their zeros are countable, and all zeros are distinct. My question ...
6
votes
2answers
508 views

Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series $$ E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}}, $$ where $z=x+iy$. We think of $E(z,s)$ as a ...
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1answer
80 views

Integrals involving trigonometric functions and polynomes

Let $P(x)$ be a real polynome. Specify all such $P(x)$ that one of the next integrals converge: $$ \int_0^{\infty} sin(P(x))dx, \int_0^{\infty} cos(P(x))dx ? $$ Among special cases are such ...
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862 views

Does the Gamma function preserve integers?

Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the ...
3
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0answers
44 views

Boersma and Glasser formula

In http://iopscience.iop.org/0305-4470/38/8/005 (A differentiation formula for spherical Bessel functions) Boersma and Glasser proved the following interesting formula ...
3
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1answer
100 views

lambert W function solution for $\ln x=a+bx^{-1}$

Is is possible to solve the equation $\ln x=a+bx^{-1}$ using the Lambert W function? I understand that the lambert W function is the solution for equations like $\ln x=bx^{-1}$, which does not apply ...
4
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1answer
164 views

Special Function, Series Expansion, or Simpler Form of a Certain Infinite Product?

$\prod _{n=1}^{\infty } \left(1+a (c+n)^b\right)$ where a > 0, b < -1, and c >= 0 Is there a special function, series expansion, or other simpler (or maybe just interesting) representation of ...
6
votes
1answer
207 views

How to prove an elementary functional equation for polylogarithms?

Let $Li_s(z)$ denote the usual polylogarithm. The elementary functional equation $$Li_{-n}(z)=(-1)^{n-1}Li_{-n}(1/z)$$ holds for $n\geq 1$. I remember only that the proof used some reproducing ...
2
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1answer
63 views

The asymptotic distribution of a subset of Bessel function zeroes

For a research problem I am working on in PDE, I need to obtain asymptotics for the counting function of $$\{0<\alpha <\lambda: \exists n\in \mathbb{N} \textrm{ such that }J_n(\alpha)=0 \textrm{ ...
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2answers
158 views

Integral involving exponential and Marcum-Q function

Do you have any suggestions to solve the following integral: $\int\limits_0^\infty {{e^{ - a{x^2}}}{Q_1}\left( {bx,cx} \right)dx}$ Thank you very much.
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1answer
150 views

Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...
12
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1answer
247 views

A hypergeometric puzzle

$$ 143\,\sqrt {3}\;{\mbox{$_2$F$_1$}\left(\frac{1}{2},\frac{1}{2};\,1;\,{\frac {3087}{8000}}\right)}= 40\,\sqrt {5}\; {\mbox{$_2$F$_1$}\left(\frac{1}{3},\frac{2}{3};\,1;\,{\frac ...
4
votes
1answer
99 views

Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients

Empirically, the Legendre functions of second kind, $Q_n(x)$, appear to be of form $$ Q_n(x)=\frac{P_n(x)}{2} \cdot\ln(\frac{1+x}{1-x})+p_n(x), $$ with $P_n(x)$ the Legendre polynomials of first kind ...
11
votes
2answers
1k views

How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...
1
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0answers
98 views

L2 norm of a M-Whittaker function

Let $M_{\kappa,\mu}(z)$ be the Whittaker function, as defined here http://en.wikipedia.org/wiki/Whittaker_function. Does any one know the evaluation of the following integral? ...