1
vote
3answers
48 views

Bounding a modified Bessel function of the first kind.

Let $I_0$ be the zeroth-order modified Bessel function of the first kind. We know that, asymptotically as $x\to \infty$, $I_0(x) \sim e^x/\sqrt{2\pi x}$. Does anybody have a reference for the maximum ...
2
votes
1answer
75 views

Looking for an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$

I have a recollection of seeing an identity connecting polylogarithm and polygamma functions of arguments $\frac14$ and $\frac34$. But I don't remember details, and searching my books and the Internet ...
0
votes
1answer
48 views

Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$ where $F(x)$ is ...
6
votes
1answer
120 views

How do people on MSE find closed-form expressions for integrals, infinite products, etc?

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I've seen some pretty complicated integrals and infinite ...
5
votes
2answers
263 views

Curious gamma identity

I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc). The identity is as follows: $$\Gamma(t) = x^t ...
37
votes
4answers
2k views

Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
0
votes
2answers
72 views

Basic question on complex integration

I have a very basic question on complex integration. How is the definite integral $$ \int_{z_1}^{z_2}{f(z)dz} $$ $z \in \Bbb{C}$ to be interpreted in the absence of a specific path over which ...
2
votes
5answers
188 views

Reference Book on Special Functions

Now I'm studying the topic that uses the special functions frequently, so I find myself in need for some good reference book on the properties and equalities of the special functions. The optimal one ...
0
votes
0answers
70 views

Learning about the gamma function.

I have just started learning about the gamma function but the books I have are not sufficient to give me a complete picture of it. Can you guys suggest some online resources/free books where I can ...
0
votes
1answer
27 views

$_2F_1\left(\frac34,\frac54,2,x^2\right)$ in terms of elliptic integrals $E$ and $K$

I am given the following formula (found to be correct numerically): $$ _2F_1\left(\frac34,\frac54,2,x^2\right) = \frac{-8}{\pi x^2} \left[ \sqrt{1+|x|}~E\left(\frac{2|x|}{1+|x|}\right)- ...
9
votes
2answers
255 views

The most complete reference for identities and special values for polylogarithm and polygamma functions

I am looking for a book, paper, web site, etc. (or several ones) containing the most complete list of identities and special values for the polylogarithm $\operatorname{Li}_s(z)$ and polygamma ...
2
votes
1answer
33 views

A Bessel related question - What is the M(u,v,phi) function?

I have came across a function, written as M(u,v,phi), where it is defined as: $$ = 1/2 \pi * \int_0^\infty e^{(u \cos(\theta)} * e^{(v \cos(2(\theta + \phi)} d(\theta) $$ To my knowledge, this ...
4
votes
1answer
92 views

What is the name for defining a new function by taking each k'th term of a power series?

With the definitions of the three functions $$ f(x)= 1 + \frac{x^3}{3!} + \frac{x^6}{6!} + ... \\ g(x)= x + \frac{x^4}{4!} + \frac{x^7}{7!} + ... \\ h(x)= \frac{x^2}{2!} + \frac{x^5}{5!} + ...
1
vote
1answer
98 views

Are Laguerre-Gaussian functions compactly supported?

Laguerre-Gaussian functions are very common in optics and I wonder if they are Compactly Supported. These functions are essentially an associated Laguerre Polynomial modulated by a gaussian function. ...
2
votes
0answers
67 views

To find the limit of three terms mean iteration

We know that the arithmetic-geometric mean $AGM(a,b)$ of $a$ and $b$ defined as $$2a_1=a+b$$ $$b^2_1=ab$$ $$2a_n=a_{n-1}+b_{n-1}$$ $$b^2_n=a_{n-1}b_{n-1}$$ $AGM(a,b)=\lim\limits_{n\to \infty} ...
8
votes
1answer
148 views

Algebraic structures whose Hilbert-Poincaré series are special functions

Are there good examples of algebraic structures whose Hilbert-Poincaré series are that of special functions? I'm particularly interested in cases where complex analytic reasoning about those series ...
4
votes
1answer
84 views

Interpolation between iterated logarithms

I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$ Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for ...
7
votes
3answers
359 views

Can you recommend some books on elliptic function?

I plan to study elliptic function. Can you recommend some books? What is the relationship between elliptic function and elliptic curve?Many thanks in advance!
2
votes
0answers
69 views

Simplification of Kampé de Fériet function

I was dealing with a convolution type integral $$ \int^z_0 t^m {}_0F_1(;1;-t) \: {}_2F_3\Big( 1,1;2,m,m+1 ; -a t\Big) \:\mathrm{d}t $$ By applying one of the identities in Exton's book, the solution ...
1
vote
1answer
46 views

Recurrence inequality for Dirichlet's eta function.

I'm studying the following function: $\theta(p)=\eta(p)\eta(p-2)-\frac{p-1}{p}\eta^2(p-1)$, where $\eta$ - Dirichlet's eta function. If we build a plot for $p\in [1,150]$, it's easy to see that it's ...
2
votes
3answers
96 views

What type of Hypergeometric series is this?

I am trying to find a closed form for the series $$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$ $m$ is a nonzero positive integer, and $b$, ...
5
votes
1answer
160 views

Integral representation of cosecant function

According to Wolfram website http://functions.wolfram.com/ElementaryFunctions/Csc/introductions/Csc/05/, There exists a "well-known" integral representation for the cosecant function, i.e. ...
2
votes
0answers
118 views

A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )

Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number. Odiuos Number Now consider the sequence of functions below $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...
0
votes
1answer
59 views

reference needed for Gamma function

Please help me to find a reference (book) for the following upper bound of Gamma function For $x \geq 1$ $$ \Gamma(x)\leq x^{x-1}. $$ Thank you.
5
votes
0answers
133 views

Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?

In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
3
votes
2answers
201 views

Proving that special functions do not have closed-form expression

When dealing with special functions, like Erf, one should encounter the following statement This function cannot be expressed in terms of classical functions This seems pretty true, but I was ...
29
votes
2answers
1k views

Is this function decreasing on $(0,1)$?

While doing some research I got stuck trying to prove that the following function is decreasing $$f(k):= k K(k) \sinh \left(\frac{\pi}{2} \frac{K(\sqrt{1-k^2})}{K(k)}\right)$$ for $k \in (0,1)$. ...
0
votes
1answer
68 views

Expansions of Hermite functions

I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients. Thank you for your ...
5
votes
0answers
113 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 ...
5
votes
0answers
138 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 ...
3
votes
2answers
300 views

What is this generalized multivariable hypergeometric function?

I'm looking for any kind of reference on a multivariable generalization of a (confluent) hypergeometric function. To be specific, Horn's List is a list of 34 two-variable hypergeometric functions, 20 ...
4
votes
3answers
354 views

Theory of the Mathieu Operator

How important is the theory of the Mathieu operator in mathematics/applied mathematics? What are the major mathematical concepts required to study it? The Mathieu operator is an ordinary periodic ...
5
votes
1answer
513 views

On the completeness of the generalized Laguerre polynomials

I am trying to prove that the generalized Laguerre polynomials form a basis in the Hilbert space $L^2(\mathbb{R})$. 1. Orthonormality \begin{equation} \int_0^{\infty} ...
6
votes
3answers
409 views

Reference requests: Jitsuro Nagura

I spent some time today looking for any biographical information on Jitsuro Nagura and came up empty-handed. Any suggestions welcome. Also, the Wiki note on the Chebyshev $\psi$ function says that ...
4
votes
1answer
466 views

Finding a generating function for the Laguerre polynomials

I've started learning some quantum physics and one often encounters special functions (like Legendre polynomials, Laguerre polynomials, Bessel functions, ...). Many calculations with these functions ...
6
votes
1answer
209 views

Laguerre polynomials and inclusion-exclusion

Does anyone know a reference for the solution of the generalized derangement problem via Laguerre polynomials? The Wikipedia article here says that this is an application of inclusion-exclusion, but ...
1
vote
1answer
224 views

Bessel's function

I read that $\int\limits_0^1 xJ_n(j_{na}x) J_n(j_{nb}x) dx={1\over 2}\delta_{ab}[J_n'(j_{na})]^2$, where $j_{na},j_{nb}$ are zeros of $J_n$, the Bessel function of the $n$th degree. Is there a ...
1
vote
1answer
141 views

What are the $\ker$, $\mathrm{kei}$ functions?

In a book titled 'Ordinary Differential Equations and Useful Polynomials', under the chapter 'Bessel's function', the author has introduced four new functions $\mathrm{ber}$, $\mathrm{bei}$, $\ker$, ...
2
votes
4answers
382 views

Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $

This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (="$\eta$") in terms of matrixoperations I asked myself, ...
1
vote
3answers
1k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
3
votes
2answers
591 views

Tables of Hypergeometric Functions

I'm looking for a book, set of tables, or other reference which contains a comprehensive list of hypergeometric identities; that is, something which allows a hypergeometric fucntion to be expressed in ...
1
vote
0answers
148 views

Orthogonal polynomial interpolation of a function

I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the ...
11
votes
1answer
502 views

Hermite's solution of the general quintic in terms of theta functions

Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...