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

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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^{...
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
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/...
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 ...
3
votes
2answers
2k views

How are the “real” spherical harmonics derived?

How were the real spherical harmonics derived? The complex spherical harmonics: $$ Y_l^m( \theta, \phi ) = K_l^m P_l^m( \cos{ \theta } ) e^{im\phi} $$ But the "real" spherical harmonics are given ...
19
votes
1answer
469 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 ...
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 ...
3
votes
0answers
167 views

Iterated Root Mean Square-Arithmetic Mean

Can I find iterated Root Mean Square-Arithmetic Mean as a function of Arithmetic-geometric mean (AGM) with some transformations if it is possible? if not possible, what is the closed form of it as ...
2
votes
1answer
192 views

Hypergeometric series

If found that : "Assume further that this equation has e series solution $\sum a_ix^i$ whose coefficients are connected by two term recurrence formula. Then, such a series can be expressed in terms ...
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}\,\...
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 ...
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^{-\...
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_\...
2
votes
2answers
209 views

a proof for the following Gamma function inequality

Could you please provide or point me to a proof of inequality 5.6.8 found at this site? That is, $\left|\frac{\Gamma(z+a)}{\Gamma(z+b)}\right| \leq \frac{1}{|z|^{b-a}}$ for $z\in \mathbb{C}$, $a,b\...
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 $b>0$.
10
votes
2answers
613 views

Prove that $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's identity.

I'm trying to derive the functional equation $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's formula: $$\Gamma(z)=\lim_{n\to\infty}\frac{n!\,\,n^z}{z(z+1)\cdots(z+n)}\,,...
9
votes
1answer
421 views

$\frac{1}{e^x-1}$, $\Gamma(s)$, $\zeta(s)$, and $x^{s-1}$

Just to give a few examples, we have that $$\eqalign{ & \int\limits_0^\infty {\frac{{{x^{s - 1}}}}{{{e^x} - 1}}dx} = \Gamma \left( s \right)\zeta \left( s \right) \cr & \int\limits_0^\...
0
votes
1answer
263 views

The ratio of two strictly increasing polynomial functions

I have the following question: Given, $f_1(a), f_2(a),\ldots, f_n(a)$ and $g_1(a), g_2(a),\ldots, g_n(a)$ are strictly increasing positive "polynomial" functions of $a$. It is also known that $$\...
1
vote
2answers
3k views

Expected value of $\ln X$ if $X$ is $\Gamma(a,b)$ distributed.

I'm new here and hope you can help. It's really late here in South Africa, maybe my mind just doesn't want to function now! But I need to figure out how to get a closed form expression hopefully for ...
4
votes
1answer
455 views

Integral with exp and erf

I found an integral calculated from what I understand with “differentation under the integration sign” method. $$ \int_{-\infty}^{\infty}\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\left(a\...
6
votes
3answers
472 views

Evaluating $ \int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x $

I have big difficulties solving the following integral: $$ \int_{-\infty}^{\infty}x\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}^{2}\left(a\left(x-d\right)\right)\,\mathrm{d}x $$ I tried to ...
1
vote
1answer
138 views

Does division of polynomials give an increasing function?

How can I show that \begin{equation} f(a)=\frac{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i+1}{\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ ...
10
votes
2answers
598 views

Ramanujan Summation

It seems that under the light of Ramanujan Summation the following is plausible: $$1 + {2^{2n - 1}} + {3^{2n - 1}} + \cdots = - \frac{{{B_{2n}}}}{{2n}}(\Re)$$ Alas, I can't really find any ...
13
votes
5answers
1k views

What are special functions for?

If you read enough mathematics, you eventually come across several so-called "special functions". I'm always left wondering what on Earth these things are actually for. We have the Euler Gamma ...
1
vote
1answer
170 views

A uniqueness proposition involving Erf, the error function

This is a MathOverflow cross-post (currently no answer there) and a generalization of a previous MathOverflow question, "Reducing system of equations involving Erf, Error Function". Consider the ...
6
votes
2answers
1k views

Asymptotic expansion of integral involving modified Bessel-function

I would like to obtain the asymptotic expression for $\alpha \to \infty$ of the following integral $$I(\alpha)=\int_0^\infty\!dx\,x (1 - \cos[2\alpha K_0(x)]) = \int_0^\infty\!dx\, 2x \sin^2[\alpha ...
6
votes
0answers
151 views

relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions

There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
2
votes
1answer
330 views

Special case of Meijer G function

I have an instance of the Meijer G function (using the definition from http://en.wikipedia.org/wiki/Meijer_G-Function, first equation there) that seems like, given its simplicity, it should be ...
0
votes
2answers
489 views

complete elliptic integral of the first kind

I'm looking for any "closed form" for the coefficient of the $\ell$-th power of $K(x)$, the complete elliptic integral of the first kind. Thanks.
7
votes
1answer
236 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $$f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$$ satisfies $$x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R.$$ How ...
3
votes
1answer
189 views

Deriving the form of the Exponential Integral from a given integral

The Wikipedia entry on Asymptotic Expansion outlines a detailed example, where it refers to the fact that the integral \begin{equation} \int_0^\infty \frac{e^{-w/t}}{1-w} \, dw \end{equation} ...
5
votes
0answers
160 views

Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?

Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
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$$\int\frac{1}{\ln x}\,dx$$$$u=\ln x,1=\frac{dx}{du}\frac{1}{x},dx=x\,du,dx=e^{\ln x}du,dx=e^u\,du$$$$\...
4
votes
0answers
154 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
2
votes
1answer
264 views

What does $ \langle Y_{lm} | Y _{\lambda\mu} \rangle = \delta_{l\lambda} \delta _{m\mu} $ mean?

In Rotation Matrices for Real Spherical Harmonics. Direct Determination by Recursion, I can almost completely understand the recurrence relations described, but for one part. The $Y^l_m$ function is ...
1
vote
0answers
152 views

Proving or disproving that if $\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$ both $a$ and $b$ are integers.

I found some formula about special function very complicated, so I am curious how you people solve this by hand. $$\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$$ but $a$ and $b$ are very ...
2
votes
2answers
6k views

Definition of Sinc function

I just want to make clear of the definition of sinc(x). I know there is a normalized and unnormalized definition for the sinc function. If we have unnormalized sinc then we have: $$\sin(x)/x=\text{...
8
votes
1answer
242 views

Can it be shown that $Y_0(\lambda_n a)J_0(\lambda_n a) - J_0(\lambda_n a)Y_0(\lambda_n a) \ne 0$?

Background I am currently looking into the task of describing a transient temperature field $\theta(r,t)$ across the thickness $a \leq r \leq b$ of an infinitely long and hollow cylinder exposed to a ...
2
votes
1answer
407 views

A Curious Binomial Coefficient Sum

Let $k, l \leq n$ be non-negative integers. Does the following identity simplify? \begin{align} \sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -l + 1}{n} = \binom{n - l + 1}{n} \phantom1_{2}\mathsf{F}...
1
vote
1answer
292 views

What is the fractional derivative of the function $\pi \cot (\pi x)$?

What is the fractional derivative of the function $\pi \cot (\pi x)$? I derived the following expression: $(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma (-p)}...
2
votes
1answer
269 views

how to evaluate $\int_0^1\sin(\frac{1}{x})dx$?

How can I evaluate the integral of $$\int_0^1\sin\left(\frac{1}{x}\right)dx.$$ Maybe it needs the cosine integral to evaluate it, but I cannot understand it very well. Thanks a lot.
4
votes
2answers
209 views

Could you give a application of a special function on number theory or analysis?

With the best effort i have ever taken, i couldn't find a application of a special function on number theory or analysis on the internet. By the way, why is the applications of special functions in ...
32
votes
2answers
2k views

Proof that $\sum\limits_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$ regarding $\zeta(3)$ and Apéry's proof

I recently printed a paper that asks to prove the "amazing" claim that for all $a_1,a_2,\dots$ $$\sum_{k=1}^\infty\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots(x+a_k)}=\frac{1}{x}$$ and thus (probably) ...
6
votes
1answer
437 views

Was the definition of $\mathrm{erf}$ changed at some point?

I have seen two competing definitions of the error function. When I was an undergrad, Spiegel's Mathematical Handbook of formulas and tables (mine is the 1968 edition) was the definitive authority, ...
2
votes
1answer
173 views

Closed form of integral of $\operatorname{erfc} \log t$

Is there any closed form expression for the following integral? $$ \int\limits_t^\infty \left(1- \operatorname{erf}(\log x) \right )dx $$ or equivalently: $$ \int\limits_t^\infty \operatorname{erfc}...
9
votes
2answers
2k views

Are there well known lower bounds for the upper incomplete gamma function?

Let $a >0, b >0$, and $r \in \mathbb{R}$. I am trying to find a lower bound for the integral $$\int_a^\infty y^{-r} \exp\left( - b(y-a)^2\right) \,\mathrm dy.$$ After consulting the Wikipedia ...