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Results tagged with special-functions
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user 56553
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.
4
votes
1
answer
144
views
Jacobi elliptic functions with modulus on the unit circle
I am gathering some available informations on Jacobi elliptic functions $sn(z,k)$, $cn(z,k)$, $dn(z,k)$ with $k\in\mathbb{C}$, $|k|=1$. I can not find much on them in standard references (Abramowitz&S …
1
vote
0
answers
113
views
Orthogonal polynomials with quadratic recurrence coefficients
Consider the monic orthogonal polynomials determined by the recurrence
$$p_{n+1}(x)=(x-n(n+b))p_{n}(x)-n(n+a)p_{n-1}(x), \quad n\in\mathbb{N},$$
with the initial conditions $p_{-1}(x)=0$ and $p_{0}(x) …
3
votes
1
answer
238
views
Challenging problems concerning Jacobian elliptic functions with complex modulus
I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since $sn(u, …
2
votes
Accepted
Challenging problems concerning Jacobian elliptic functions with complex modulus
The conjecture has been verified. For the proof and other interesting details, see http://arxiv.org/abs/1512.06089.
2
votes
1
answer
203
views
An extreme of Jacobi elliptic function on an interval
Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the this func …
1
vote
0
answers
102
views
Asymptotic behavior of hypergeometric function ${}_{3}F_{2}(a,b-n,c-n;d-n,e-n;1)$ for $n\to\...
Suppose $a,b,c,d,e\in\mathbb{R}$ are such that $d+e-a-b-c>0$ and $d,e\notin\mathbb{Z}$. I would like to know whether it is possible to deduce an asymptotic formula for the sequence given by the hyperg …
7
votes
1
answer
306
views
Asymptotic behavior of a sequence of functions
For $n\in\mathbb{N}$ and $q\in(0,1)$, define
$$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ …
4
votes
0
answers
239
views
A connection between basic hypergeometric series and number theory
I am studying functions given by the power series:
$$f(z)=1+\sum_{n=1}^{\infty}\frac{z^n}{(1-q)(1-q^2)\cdots(1-q^{n})}.$$
The parameter $q$ is usually assumed to be such that $|q|<1$. Then it is easy …
6
votes
1
answer
353
views
Asymptotic behaviour of an integral
For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define
$$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$
where
$$g(\theta)=\int_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1 …
9
votes
3
answers
404
views
An explicit representation for polynomials generated by a power of $x/\sin(x)$
The coefficients $d_{k}(n)$ given by the power series
$$\left(\frac{x}{\sin x}\right)^{n}=\sum_{k=0}^{\infty}d_{k}(n)\frac{x^{2k}}{(2k)!}$$
are polynomials in $n$ of degree $k$. First few examples:
$ …