**3**

votes

**2**answers

90 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**

votes

**1**answer

394 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 ...

**0**

votes

**2**answers

195 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 $0\lt q\lt 1$ ?

**9**

votes

**2**answers

344 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
...

**4**

votes

**2**answers

110 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

630 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 ...

**2**

votes

**1**answer

102 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 ...

**5**

votes

**0**answers

106 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 ...

**30**

votes

**1**answer

1k 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 ...

**0**

votes

**0**answers

51 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

129 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

218 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

223 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

224 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

132 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

449 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

155 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

34 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

210 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

240 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

82 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

65 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

73 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

856 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

152 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

37 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

80 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

96 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 ...

**5**

votes

**1**answer

184 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

59 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{ ...

**1**

vote

**1**answer

82 views

### What function is “$U_{\nu}(\cdot, \cdot)$”?

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer.
Integral 1.8.2.4:
$$
\int_0^x x^{\nu+1} ...

**1**

vote

**2**answers

102 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.

**0**

votes

**1**answer

145 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, ...

**4**

votes

**1**answer

79 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 ...

**5**

votes

**2**answers

697 views

### On a polynomial related to the Legendre function of the second kind

The Legendre function of the second kind, $Q_n(z)$, along with the usual Legendre polynomial $P_n(z)$, are the two linearly independent solutions of the Legendre differential equation.
$Q_n(z)$ can ...

**11**

votes

**2**answers

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 ...

**23**

votes

**6**answers

3k views

### Why are hypergeometric series important and do they have a geometric or heuristic motivation?

Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia ...

**1**

vote

**0**answers

63 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?
...

**2**

votes

**3**answers

390 views

### About Turan`s problem(inequality) in multivariable

Hi. I have a question related to Turan`s problem, that is
Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$.
I am considering the generalized question for ...

**2**

votes

**2**answers

153 views

### Estimate of a ratio of two incomplete gamma functions

I would like to bound from above the expression
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}
$$
for $x>y>0$. By plotting the above expression I have found that ...

**0**

votes

**0**answers

65 views

### A solution of a q-difference equation

Is it possible to find a solution of the $q$-difference equation
$$f(q^{-1}x)-f(qx)=x(a-x)f(x),$$
with $f(0)=1$, (perhaps) in terms of basic hypergeometric series? Or in another rather explicit form? ...

**3**

votes

**2**answers

130 views

### A calculation involving Lerch Transcendents

The Lerch Transcendent is defined here as
$$\Phi(z,s,a):=\sum_{k=0}^\infty \frac{z^k}{(k+a)^s}.$$
I am interested in the case $z=\frac 12,$ $s=1.$ The following limit showed up in estimating uniform ...

**2**

votes

**1**answer

89 views

### M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...

**1**

vote

**2**answers

131 views

### Evaluate an integral or Fourier coefficients

Consider an integral
$$
\int_0^\pi \frac{\cos(kx)}{\cosh(ax)}\ dx
$$
there $k\in
\mathbb{Z}, a\in \mathbb{R}.$
Of course that is Fourier coefficient for the function $f(x)=\frac{1}{\cosh(ax)}.$
...

**0**

votes

**0**answers

82 views

### Integral of Bessel function of 1st kind with complex exponential

Does someone know the solution (simple closed form) of one of theses integrals:
$$\int_0^t J_l(s) e^{-iA(t-s)}ds$$
$$\int_0^t \frac{J_l(s)}{s} e^{-iA(t-s)}ds$$
with $l>0$, $t>0$, $\Re(A)>0$, ...

**6**

votes

**1**answer

172 views

### Abel's five terms relation from Yang-Baxter equation?

Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations?
If yes, how?
Thanks for any help.

**3**

votes

**1**answer

194 views

### Computing Reciprocal Gamma

Reciprocal Gamma $1/\Gamma(z)$ is an entire function and so it has a convergent Taylor series expansion which was given in its wikipedia article.
...

**3**

votes

**3**answers

445 views

### A “known” tangent half-angle formula?

In another posting I wrote about a trigonometric relation I had derived, but that ended up not being the main point of the posting:
Strange pattern in rounding errors?
So as long as we're here, ...