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

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4
votes
1answer
196 views

Infinite series representation. Limited or not?

Recently I've found the following. Let $n < m$. Then the integral $$\int\limits_0^\infty {\frac{{{x^{n-1}}}}{{1 + {x^m}}}} dx$$ converges and its value is (using the $B$ and $\Gamma$ function ...
10
votes
1answer
594 views

Evaluate or simplify $\int\frac{1}{\ln x}\,dx$

I did a bit of work on this, but I'm not so sure about the parts towards the end. Starting with$$\int\frac{1}{\ln x}\,dx$$$$u=\ln x,1=\frac{dx}{du}\frac{1}{x},dx=x\,du,dx=e^{\ln x}du,dx=e^u\,du$$$$\...
2
votes
0answers
68 views

Is this formula for $ \sum_{n} (n^{2}+z^{2})^{-s} $ correct?

I would like to know if this formula is true: $$\sum_{n=1}^{\infty}\frac{1}{(z^{2}+n^{2})^s}=\frac{1}{\Gamma(s)} \sum_{n=0}^{\infty}\Gamma(s+n)\zeta(2s+2n)\frac{ (-z^2)^n}{n!}.$$ I have used the ...
5
votes
2answers
157 views

Riemann zeta sums and harmonic numbers

Given the nth harmonic number of order s, $$H_n(s) =\sum_{m=1}^n \frac{1}{m^s}$$ It can be empirically observed that, for $s > 2$, then, $$\sum_{n=1}^\infty\Big[\zeta(s)-H_n(s)\Big] = \zeta(s-1)-...
6
votes
1answer
300 views

On the Dirichlet beta function sum $\sum_{k=2}^\infty\Big[1-\beta(k) \Big]$

Given the Dirichlet beta function, $$\beta(k) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^k}$$ (The cases k = 2 is Catalan's constant.) It seems, $$\sum_{k=2}^\infty\Big[1-\beta(k) \Big] = \frac{1}{4}\...
1
vote
0answers
85 views

integral with bessel function represented as a series [duplicate]

Possible Duplicate: prove equality with integral and series This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
2
votes
1answer
116 views

To simplify $f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$

Let $x\leq0$, then $$ f_a(x)= \int_{-a}^{+a} e^ {-\frac{x}{t^2-a^2}}\;dt$$ $$ f'(x)= -\int_{-a}^{+a} \frac{1}{t^2-a^2} e^ {-\frac{x}{t^2-a^2}}dt$$ $$ f'(x)= -\int_{-a}^{+a} \frac{t^2-(t^2-a^...
1
vote
0answers
134 views

How to find a function with the following properties?

I want to find a function $f(s,x)$ such that $f(s,x)$ is analytic for any $s \in Z^+ $, $f(s,x)=B_s(x)$, where $B_s(x)$ are the Bernoulli polynomials $f(a, x)$ is elementary against $x$ at any ...
5
votes
1answer
227 views

prove equality with integral and series

I am stuck on one question with integral. Help me please to show that with $n=1$ the following is true $$ \int_{0}^{\infty}\left(\frac{2^n}{t^n}\left(\frac{t^n}{2^nn!}-\frac{1}{2^{n+2}}\frac{t^{n+2}}{...
4
votes
1answer
143 views

Hypergeometric functions inequality

Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers. From a simple plot it looks like $_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} \,_2F_1(m+n,1,n+1,\...
4
votes
2answers
259 views

Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
16
votes
4answers
508 views

The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$

Does anyone know if this function has a name? I came up with it by looking at the power series for $e^z$, changing the summation to an integral, and substituting the gamma function for the factorial ...
1
vote
0answers
716 views

solving Bessel function equation by hand

I have a Bessel function of the first kind given by the equation $$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m+\alpha}$$ I am trying to ...
15
votes
1answer
369 views

On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others

For p = 2, we have, $\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$ It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then, ...
11
votes
1answer
224 views

A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation $$\frac{\partial}{...
2
votes
0answers
181 views

Can we confirm $\sum_x{x!}$?

According to Wikipedia's article on indefinite sums, they list the following formula near the bottom of the page: $$\displaystyle \sum_x{\Gamma(x)}=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}{e}+C$$ ...
0
votes
1answer
173 views

How to determinate the linearly independence between some special functions defined by ODE?

How to determinate the linearly independence between some special functions defined by ODE? For example: ${}_1F_1(a;b;x)$ , $x^{1-b}{}_1F_1(a-b+1;2-b;x)$ when $b$ is integer ${}_2F_1(a,b;c;x)$ , $x^{...
0
votes
1answer
96 views

Condition for frame of $L_2$

Let $f$ be continuous, real valued and compactly supported with exactly one maximum function in $L_2$. Form the functions $$ f_{m,k}=f^m(x-2^k) $$ Under which conditions $\{f_{m,k}\}$ would be a ...
2
votes
2answers
397 views

Polynomials in Fourier trigonometric series

I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding: $\frac{\sin{kx}}{k}$, $\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$, $\frac{2 x \cos{kx}}{k^...
10
votes
3answers
324 views

Nicer expression for the following differential operator

I have the following sequence of differential operators: $$D_n = \underbrace{t \partial_t t \partial_t \dots t \partial_t}_{\text{$n$ times}}.$$ Is there any expression involving a sum of "normal" ...
7
votes
0answers
160 views

Evaluting $ \int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

While working on mixture (variance) of normal distribution and keep running into these two integrals $$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$ $...
5
votes
2answers
229 views

What is the value of $\Gamma(\mathrm{i})$ ?

What is the value of $\Gamma(\mathrm{i})$ ? $\Gamma(z)$ is Gamma function. Here $\mathrm{i}^2=-1$.Can you help me with this problem ?
0
votes
1answer
105 views

Is there a closed form for $\int_{-\infty}^{\infty} \frac{e^{-ax^2}}{\mathrm{erfc}{(-bx)}} dx$?

The integral expression is $$ I = \int_{-\infty}^{\infty} \frac{e^{-ax^2}}{\mathrm{erfc}{(-bx)}} dx $$ where $a>0$ and $b>0$.
4
votes
1answer
117 views

Beta integral transformation

It's a homework task and I can't get past the last step. Task is to prove that $$ B(x,y)=\int\limits_0^1 \frac{\tau^{x-1}+\tau^{y-1}}{(1+\tau)^{x+y}} \mathrm{d}\tau $$ By substituting $t=\frac{1}{\...
2
votes
0answers
74 views

Lower bound for the eigenvalue

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
2
votes
0answers
175 views

Hermite functions and integral

Let $$ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $$ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function. Consider $$ k_n(x,y)=\frac{h_{n+1}(x)h_n(y)-h_{n+1}(y)h_n(x)}{...
2
votes
2answers
200 views

Equality involving Appell hypergeometric function

After some algebra, Wolfram online integrator gave me the following: $$\tag{1} \int (1-a-t)^{N-2}\ \sqrt{2t-t^2}\ \text{d} t = c\ \cdot t^{3/2}\ \operatorname{F}_1 \left( \frac{3}{2}; -\frac{1}{2}, 2-...
1
vote
0answers
90 views

$n$-th derivative of the prolate spheroidal function

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
5
votes
1answer
402 views

How to prove Gauss's Digamma Theorem?

Here $\psi(z)$ is digamma function, $\Gamma(z)$ is gamma function. $$\psi(z)=\frac{{\Gamma}'(z)}{\Gamma(z)},$$ For positive integers $m$ and $k$ (with $m < k$), the digamma function may be ...
5
votes
2answers
164 views

Procedure for evaluating the hypergeometric series $_2F_1\left\{\frac{v+2}{2},\frac{v+3}{2};v+1;z\right\}$

I'm trying to work out the procedure to get the following hypergeometric series into a simpler form, for all postive integer $v$: $$ _2F_1\left\{\frac{v+2}{2},\frac{v+3}{2};v+1;z\right\}$$ For ...
2
votes
0answers
224 views

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference http://bigwww.epfl.ch/publications/...
1
vote
0answers
297 views

Using Rouche's theorem

Let $p>1$. Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$. Function $\phi(p)$ is analytic on its domain. It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\...
4
votes
2answers
506 views

Finding some orthogonality in a convolution-like integral over Legendre polynomials

I encountered the following integral in my (physics) research, and I've yet to find an analytic solution: $$I(n_1,n_2,n_3) = \int_{-1}^{1} d(\cos\theta_1) \int_{-1}^{1} d(\cos\theta_2) P_{n_1}(\cos\...
2
votes
2answers
263 views

Relating Gamma and factorial function for non-integer values.

We have $$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$ for integers, so if $\Delta$ is some real value with $$0<\Delta<1,$$ then $$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$ because $...
2
votes
1answer
127 views

Is this series expressible in terms of Gauss' hypergeometric function?

How we can express this series $$F(z)=\sum_{n=0}^\infty \frac{z^n}{(a)_nn!}$$ in terms of Gauss' hypergeometric function? where $(a)_n$ denotes the Pochhammer symbol. Thanks in advance
4
votes
2answers
1k views

Why are the Gegenbauer polynomials called “ultraspherical”?

There has to be a good reason why the Gegenbauer polynomials were also named "ultraspherical" polynomials. I am aware that when $\alpha=\frac{1}{2}$, the Gegenbauer polynomials reduce to the Legendre ...
19
votes
1answer
468 views

What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

We have the following evaluations: $$\begin{aligned} &\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = \frac{1}{3}\,\...
2
votes
1answer
149 views

What representation should I choose for numerical computation of hypergeometric function ${}_2 F_1(1+i\eta, 2; 2+i\eta; x)$ where $|x|=1$

I have a task - to plot graphics of the function: $$ I(E) = \frac{16i \pi k \mu}{(\beta - ik)^{4}} \frac{1}{1 + i\eta} {}_2 F_1(1+i\eta, 2; 2 + i \eta; x) $$ where $$ x = \left( \frac{\beta + ik}{\...
9
votes
1answer
591 views

Original author of an exponential generating function for the Bernoulli numbers?

The Bernoulli numbers were being used long before Bernoulli wrote about them, but according to Wikipedia, "The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of ...
0
votes
1answer
134 views

Are there names for the indices of the spherical harmonics?

I know that physicists call $\ell$ and $m$ the "azimuthal" and "magnetic" quantum numbers, respectively. But those sound very physics-y. (I am actually a physicist, but still.) Are there names for ...
7
votes
1answer
823 views

Physical interpretation of the generating function for the Bessel functions.

It is well known that the generating function for the Bessel function is $$f(z) = \exp \left (\frac12 \left (z - \frac1z \right ) w \right ).$$ So, we have $$f(z) = \sum_{\nu = -\infty}^{\infty} J_\...
4
votes
0answers
105 views

analytic evaluation of $\int_{0}^{\infty} \frac{dx}{\mathrm{Bi}(x)}$

So, just out of random curiosity, I'm trying to find an analytic expression for the following definite integral: $$\int_{0}^{\infty} \frac{dx}{\mathrm{Bi}(x)}$$. Where $\mathrm{Bi}(x)$ is the Airy ...
6
votes
2answers
163 views

Direct evaluation of complete elliptic integral

In comments to this question, @RobertIsrael asserted that, for $-1<x<1$, $$ \int_0^{2\pi} \frac{1-x \cos(\phi)}{\left(1 - 2 x \cos(\phi) + x^2\right)^{3/2}} \mathrm{d} \phi = \frac{4}{1-x^2} ...
2
votes
0answers
914 views

How to prove that Legendre polynomials form a complete basis using functional analysis

I would like to prove that the Legendre polynomials form a complete basis on the interval [-1, 1] using functional analysis. Here is what I came up with so far. Legendre polynomials $P_n(x)$ are ...
5
votes
2answers
220 views

A typo in a formula of Ramanujan?

In Mathworld's article Gamma function, in line (96), we find the formula, $\sum_{k=0}^\infty (8k+1)\left(\frac{\Gamma(k+\frac{1}{4})}{k!\;\Gamma(\frac{1}{4})}\right)^4 = 2^{3/2}\frac{1}{\sqrt{\pi}\,\...
7
votes
1answer
1k views

Bessel function integral and Mellin transform

Gradshteyn&Ryzhik 6.635.3 provides the following integral, with the usual constraints on $\nu,\alpha,\beta$, $$\int\limits_0^\infty \exp\left(-\frac{\alpha}{x}-\beta x\right)J_\nu(\gamma x)\frac{\...
6
votes
1answer
1k views

Some questions about the gamma function

Show that $\Gamma(y) = \int_0^{\infty}{e^{-x}x^{y-1}\,dx}$ is finite for $y>0$ both as an improper Riemann integral and as a Lebesgue integral. Show $\Gamma'(y) = \int_0^{\infty}{e^{-x}x^{y-1}\...
4
votes
1answer
146 views

Convergence of this integral [duplicate]

Possible Duplicate: Some questions about the gamma function My statistics text book prescribed by my school states that the integral $$\Gamma(n)=\int_{0}^{\infty}e^{-x}x^{n-1}dx$$ is ...
6
votes
1answer
3k views

Connection between Legendre polynomial and Bessel function

In Abramovitz and Stegun (Eq. 9.1.71) I found this curious relation $$\lim_{\nu\to\infty} \left[ \nu^\mu P_\nu^{-\mu}\left(\cos \frac{x}{\nu} \right) \right]= J_\mu(x) \qquad(1)$$ valid for $x>0$. ...
2
votes
1answer
200 views

Limit of the function $\zeta(x)/\zeta(x+1)$ as $x \to \infty$

I am looking for a simple proof that $\zeta(\alpha)/\zeta(\alpha+1) \to 1$ as $\alpha \to \infty$ (where $\zeta(\alpha)$ denotes the Riemann zeta function, $\zeta(\alpha) = \sum \limits_{n\geq 1} n^{-\...