# Tagged Questions

**5**

votes

**0**answers

110 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 ...

**3**

votes

**0**answers

97 views

+50

### Asymptotic behavior of a sequence of functions

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

**2**

votes

**1**answer

294 views

### monotonicity of functions related to modified Bessel function

Dear colleagues,
I recently met some problems related to the modified Bessel funtions of the first kind and the second kind. I want to know if there exist some results on the monotonicity of
...

**11**

votes

**2**answers

376 views

### On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation
...

**0**

votes

**1**answer

57 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 ...

**2**

votes

**1**answer

135 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 ...

**34**

votes

**1**answer

2k views

### Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...

**1**

vote

**0**answers

163 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 ...

**1**

vote

**2**answers

132 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 $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$

**5**

votes

**1**answer

455 views

### Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...

**1**

vote

**0**answers

84 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 ...

**1**

vote

**1**answer

141 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**

vote

**0**answers

100 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}] ...

**1**

vote

**1**answer

91 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!

**1**

vote

**0**answers

66 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**

votes

**2**answers

124 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**

votes

**1**answer

446 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.
...

**7**

votes

**3**answers

1k views

### a dilogarithm identity: known or new?

I was playing around with dilogarithms and numerically found the following dilogarithm identity:
$$\text{Li}_2\left(\frac{2 m}{m^2+m-\sqrt{((m-3) m+1)
\left(m^2+m+1\right)}-1}\right)$$
...

**0**

votes

**1**answer

82 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 $\rho,s>1$, ...

**6**

votes

**0**answers

314 views

### Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as,
$$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$
and the Dedekind eta function,
$$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} ...

**5**

votes

**0**answers

195 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 $n$
...

**9**

votes

**8**answers

1k views

### A good reference to grok hypergeometric functions?

When I was introduced during my degree to special functions, I made friends with a number of nice functions - Laguerre, Legendre, Hermite, Bessel, and whatnot - but I made only a passing acquaintance ...

**3**

votes

**0**answers

153 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 ...

**0**

votes

**2**answers

197 views

### A certain sum with q by the power of binomial (n 2)

Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n-1)/2}$
for all $a>0$ and $0\lt q\lt 1$ ?

**4**

votes

**2**answers

125 views

### Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$

Generally, hypergeometric function ${}_2F_1[a,b,c;z]$ is defined using Gauss series ${}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n$ with $|z|<1$, and there seems to be a lot of ...

**19**

votes

**3**answers

639 views

### Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...

**5**

votes

**0**answers

117 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 ...

**1**

vote

**0**answers

85 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 > ...

**31**

votes

**5**answers

3k views

### Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...

**2**

votes

**1**answer

138 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: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$
...

**5**

votes

**2**answers

243 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 ...

**2**

votes

**2**answers

231 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} ...

**12**

votes

**1**answer

238 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 ...

**3**

votes

**0**answers

136 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 ...

**4**

votes

**3**answers

2k views

### Finding a recursion for a sum of Legendre polynomials

The polynomial
$a_n(x):=P_n(x)-\frac{n-1}{n}P_{n-2}(x)$
where $P_n(x)$ is a Legendre polynomial came up while I was investigating methods for estimating the error in Gaussian quadrature.
I am ...

**6**

votes

**2**answers

483 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 ...

**4**

votes

**1**answer

182 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 ...

**1**

vote

**0**answers

37 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 ...

**6**

votes

**1**answer

218 views

### Logarithm of the hypergeometric function

For $F(x)={}_2F_1 (a,b;c;x)$, with $c=a+b$, $a>0$, $b>0$, it has been proved in [1] that $\log F(x)$ is convex on $(0,1)$.
I numerically checked that with a variety of $a,\ b$ values, $\log ...

**4**

votes

**2**answers

243 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 ...

**4**

votes

**1**answer

86 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
$$
...

**1**

vote

**1**answer

73 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 ...

**1**

vote

**1**answer

78 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 ...

**7**

votes

**2**answers

861 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 ...

**0**

votes

**1**answer

164 views

### Legendre differential equation with additional term

In an application I encountered the ODE
$$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f
\left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x
\right) \right) \left( ...

**3**

votes

**0**answers

42 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**

votes

**1**answer

95 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 ...

**1**

vote

**1**answer

105 views

### Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)?
That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain ...

**6**

votes

**1**answer

195 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**

votes

**1**answer

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{ ...