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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242 views

A coincidence concerning Fermat primes, binomial sums, and eta quotients?

Let $w_k$ be a primitive k th root of unity, where k is a power of 2. In response to a question, Robert Israel gave the solution, $$\sum_{n=0}^\infty \frac{(-1)^n}{\binom{kn}{kn/2}} = ...
11
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1answer
395 views

Schur functors generalization to “Jack”, “Hall-Littlewood”, “Macdonald” functors ?

Schur functors are functors from the category of vector spaces to itself. If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get ...
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0answers
485 views

Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?

I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...
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0answers
930 views

An inverse Laplace transform involving Error function

Dear MOs, I need to calculate the inverse Laplace transform of the following function $$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$ I have checked, among many ...
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1answer
177 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 $0\lt q\lt 1$ ?
3
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1answer
582 views

Fourier and Bessel

Oliver Heaviside, on page 387 of Electrical Papers, Vol. I, Macmillan and Co., 1892, available here, writes $$v = 1 - \frac{n^2r^2}{2^2} + \frac{n^4r^4}{2^2 4^2} - \frac{n^6r^6}{2^24^26^2} + \ldots ...
28
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1answer
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 ...
7
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4answers
457 views

Trig functions based on convex curves

Pardon my naivety, but I wonder if much use has been found for trigonometric functions defined in terms of a centrally symmetric convex curve $K$ replacing the circle $C$. For example, here is the ...
4
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1answer
607 views

q-Pochhammer Symbol Identity

Is this identity or an equivalent one already referenced in the litterature? In particular, is it even true? ${\frac{\left ( -1 ; e^{-4\pi} \right) ^2_{\infty}}{\left ( e^{-2\pi} ; e^{-2\pi} \right) ...
4
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3answers
508 views

Monotonicity of a combination of Bessel functions

Prove that the following function is decreasing (as a function of a) for a > 0 when 0 < r < 1: $${K_2(ar)I_2(a)-I_2(ar)K_2(a)\over I_2(a)}I_2(ar).$$ The problem arose in the analysis of a model ...
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1answer
217 views

A uniqueness proposition involving Erf, the error function

This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function". Consider the system of equations: $$1/2 + {\rm Erf}(x) - \alpha {\rm ...
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votes
1answer
393 views

Sum over Hypergeometric function 2F1 (generating function)

Dear mathematicians, in my current research project I came accross this very bothersome sum over a rather simple hypergeometric function, or formulated differently: a sum over squared binomial ...
4
votes
2answers
606 views

Reducing system of equations involving Erf, Error Function

I have a system of equations: $$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$ $$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$ Where $x \le y$ and ${\rm Erf}$ is the Error Function. By ...
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2answers
842 views

Is there a known formula for fractional derivative of cot x?

I wonder if there any established formula for fractional derivative of a function $\pi \cot (\pi x)$. I derived the following expression: $(\pi \cot (\pi ...
2
votes
1answer
230 views

Infinite Series of 2F1

I will be grateful for any ideas to solve the series $$\sum^\infty_{k=0}\frac{x^k z^k}{k!} \frac{\Gamma(1+a+2k)}{\Gamma(2+k)}{}_2F_1(1,1+a+2k;2+k;z)$$ $a$ is a nonegative integer, $z$ and $x$ are ...
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0answers
731 views

Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ...
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2answers
1k views

Duality of eta product identities: a new idea?

Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs: let's call two eta product identities $\sum\limits_{i=1}^r ...
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2answers
295 views

Elliptic function with constant real part on the unit square diagonals?

Consider the following even meromorphic doubly periodic function with poles at the gaussian integer lattice. $H(z) = \prod_{n \in \mathbb{Z}} {1 \over{ 1 - {1 ...
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0answers
308 views

Linear eta product identities - how many are there?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n}) $, let for brevity $e_k:=\eta(q^k)$. With this notation, a blog entry of Michael ...
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1answer
230 views

derivative of a special function in integral form

What is the derivative of $Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e, $$\frac{\partial}{\partial x}Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right), \quad ...
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0answers
272 views

About a Christoffel-Darboux-type sum

Hi! I've been using the Christoffel-Darboux identity for the Hermite polynomials, $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^n n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$ for some ...
3
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1answer
249 views

The relationship between Stirling number of the second kind and the polylogarithm

It is shown here on Mathworld's page on Stirling number of the second kind that $$ \sum_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}_{1-n}(1+\frac{1}{z}) $$ where $S(n,k)$ is Stirling number of the ...
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179 views

Convert 2F1 to polynomial

Is there any transformation to convert each of the following versions of ${}_2F_1$ to a polynomial? The first one is $${}_2F_1\left(\frac{1-a}{2}, -\frac{a}{2}; b;\frac{4z}{(1+z)^2} \right), \quad ...
5
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1answer
459 views

Contour Integral with Gamma functions and 2F1

Given the following contour integral $$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\, {}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: ...
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1answer
450 views

Quadratic Transformation of the Hypergeometric Function 2F1

The function ${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)$ can be transformed (as reported by A. Erdélyi) by the following formula ${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)= ...
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1answer
181 views

About one series. Are there some related special functions?

Hello, I have the following series: $$ \sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0, $$ where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum ...
4
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1answer
876 views

minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p

Take an odd prime $p$ and put $x_0:=\sum\limits_{j=0}^{p-1}\left(a_{j}\sqrt{p}\cos\dfrac{j\pi}p+b_{j}\sin\dfrac{j\pi}p +c_{j}\tan\dfrac{j\pi}p\right)$, where the $a_{ij}$ are integers. If $f$ denotes ...
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665 views

Cubic polynomials with “nice” roots, which can be expressed by trig functions of rational angles

Consider the cubic polynomial $x^3-ax+b$ for $a,b\in\mathbb N$. It has three real roots which, by Cardano's formula, can of course be written in closed form using thirds of angles or cube roots of ...
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2answers
825 views

Which trigonometric identities involve trigonometric functions?

Another question that's getting no answers on stackexchange: Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled ...
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2answers
852 views

a dilogarithm identity: known or new?

I was playing around with dilogarithms and numerically found the following dilogarithm identity: $$\text{Li}_2\left(\frac{2 m}{m^2+m-\sqrt{((m-3) m+1) \left(m^2+m+1\right)}-1}\right)$$ ...
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0answers
260 views

Finite sum of modified bessel function of the second type

I have a sum of the form $\sum^n_{i=0} \frac{{t}^{i} z^{i}}{i!}K_{a+i}(z), \quad\quad z\in\mathbb{R},z>0$ where $K$ is the modified Bessel function of the second type and $a$ is an integer, ...
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2answers
636 views

Inverse Hankel Transform

I was reading through Akhiezer's book Lectures on Integral Transforms and in chapter nine, he states that the Hankel transform is unitary for $\nu > -1$, so that for a suitable function, $f$, ...
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1answer
234 views

is this bound on a kummer function known?

Is it already known that ${}_1F_1(a;b;x) \leq \Gamma(b)(1+|x|)^{-a}$ when $a$ is an integer, $a <0,$ and $b>0?$ If it is, what is a reference? my proof: Since the Kummer function can be ...
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1answer
315 views

Upper bounds on generalized Laguerre polynomials

I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable. Are there any known simple (e.g. exponential) upper bounds on the ...
3
votes
1answer
503 views

Generalized trigonometric functions $Cos(n) v$ and $Sin(n) v$.

I just discovered a paper from 1948, Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen by R. Grammel which introduces functions he calls Cos(n) and Sin(n), representing a parameterization of the ...
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1answer
529 views

infinite series with Hypergeometric functions

Can we get a closed form for the series $\sum^\infty_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}_2F_1(k+a,k+\frac{1}{2};n+1,x)$ any hints or clues are welcomed.
5
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1answer
979 views

Two-variable generating functions for Laguerre polynomials

Where can I find generating functions for orthogonal polynomials in two variables? Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for $$ \sum_{n=0}^\infty ...
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2answers
1k views

How to do integrals involving two Bessel functions and another function?

I often encounter the integrals in the following form: $\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$, where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...
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754 views

Bessel functions in wave propagation and scattering

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small ...
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3answers
551 views

Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$

I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression $ \sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z] $ This is not very helpful though, and I 'd ...
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3answers
707 views

How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$ $f:\mathbb{R} \rightarrow \mathbb{R}$ $g:\mathbb{R} \rightarrow \mathbb{R}$ For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below? $f(x+a)=f(x)+a\times ...
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2answers
2k views

Sum of trigonometric functions

Do somebody know the closed form of the following sum (m is an integer) $$f(\beta)=\sum _{k=1}^{2m+1} \sin^{2 m+1}\left[\frac{-\beta+k \pi }{1+2 m}\right]$$ If instead of $n=1+2m$ we put $n=2m$, ...
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396 views

An infinte series involving the Modified Bessel Function of the second kind

The following series has had me held up for the past one week: $$ \sum_{n=0}^\infty\frac{(2m)_n m^n}{(2m+1/2)_n n!}A^{3n/2} t^n K_{2m+n-1/2}(2\sqrt{A}t)~~~~ A>0, ~t>0, ~m\geq1/2 $$ where ...
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429 views

modified bessel fucntion of the third kind

Hi I'm doing a computation where the modified bessel function of the third kind is the main source of computational strain, we are using a 10,000 dimension's for our distribution, is there any easier ...
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1answer
439 views

Legendre Polynomial Identity

I have encountered the following sum involving Legendre polynomials, which I hope to reduce to something involving a $\delta$-function: $$ \frac{d^2}{dx^2} \sum_{\ell=0}^{\infty} \frac{2 \ell + 1}{2 ...
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2answers
676 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 ...
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3answers
472 views

Differential equation with some constraints

I posted this to stackexchange, and after some hours got a comment that was so pessimistic about finding some neat orderly solution, that I'm posting it here too. (In case anyone cares, this is ...
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0answers
249 views

High dimensional beta integral (question following the previous post)

Hello, This post is a question following the previous post. In one dimensional case, we have $$ \int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} ...
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2answers
439 views

High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello, When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake: $$ \int_{R^n} |x-y|^{-n+\alpha} ...
11
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1answer
2k views

Geometric meaning of a trigonometric identity

It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then $$ a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta ...