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        Questions tagged [special-functions]

        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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        323 views

        Are these 5 the only eta quotients that parameterize $x^2+y^2 = 1$?

        Given the Dedekind eta function $\eta(\tau)$, define, $$\alpha(\tau) =\frac{\sqrt2\,\eta(\tau)\,\eta^2(4\tau)}{\eta^3(2\tau)}$$ $$\beta(\tau) =\frac{\eta^2(\tau)\,\eta(4\tau)}{\eta^3(2\tau)}\quad\;$$...
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        1answer
        105 views

        Eigenvalues Sturm-Liouville Operator

        Is the eigenvalue decomposition of the Sturm-Liouville operator $$ Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0, $$ with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\...
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        Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

        $\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$ where $k\in\mathbb Q$ and $p$ is a ...
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        votes
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        149 views

        “One half of a theta-function” - is there something in the literature about it?

        In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) I encountered the series $$ F(t,q):=\sum_{n=1}^\...
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        1answer
        125 views

        Time ordered integral involving beta function:

        Any help on unpacking integrals of the following type, would be helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$. Edit/...
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        37 views

        Finding the inverse of an incomplete beta function

        Is there a rigorous way of inverting $$\rho(r)=\frac{b_{0}}{1-q}B\left(1-\left(\frac{b_{0}}{r}\right)^{1-q},\frac{1}{2},\frac{1}{q-1}\right)$$ where $B\left(1-\left(\frac{b_{0}}{r}\right)^{1-q},\...
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        33 views

        Connection Problem for the Confluent Heun Equation

        Consider the Confluent Heun Equation (CHE) written in its non-symmetrical canonical form, i.e, $$y''(z)+\left(4p+\frac{\gamma}{z}+\frac{\delta}{z-1}\right)y'(z)+\left(\frac{4p\alpha z-\sigma}{z(z-1)}\...
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        votes
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        131 views

        Mirror site for the NIST Digital Library of Mathematical Functions (DLMF)

        For research I use the NIST Digital Library of Mathematical Functions (https://dlmf.nist.gov/) quite often for looking up basic facts about special functions. At the moment, however, this gives ...
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        vote
        1answer
        94 views

        Correction terms in the asymptotic expansion of hypergeometric function

        I am interested in obtaining the asymptotic expansion of $r(\rho)$ $($which is the inverse of $\rho$ below$)$, $$\rho=\frac{2b}{1-q}\left(1-\left(\frac br\right)^{1-q}\right)^{1/2}\left(_2F_1\left(\...
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        votes
        0answers
        173 views

        On the determinant $\det[\sec2\pi\frac{jk}p]_{0\le j,k\le(p-1)/2}$

        On the basis of my computation, I have the following conjecture involving the secant function. Conjecture. Let $p$ be an odd prime and define $$S_p:=\det\left[\sec2\pi\frac{jk}p\right]_{0\le j,k\le (...
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        votes
        2answers
        587 views

        A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?

        I have the following conjecture involving a possible new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$ with $p$ an odd prime. Conjecture. Let $p$ be an odd ...
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        votes
        1answer
        308 views

        A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$

        On the basis of my computation, here I pose my following conjecture involving the cosine function. Conjecture. For any positive integer $n$, we have the identity $$\frac1{2n}\det\left[\cos\pi\frac{jk}...
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        votes
        1answer
        110 views

        On the convergence of MIller's Algorithm for special function evaluation (hypergeometric 1F1)

        This is going to a longish question, so the short version first: Is there a way to sanity-check which solution to the 3-term recurrence relation an application of Miller's algorithm has converged on? ...
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        votes
        1answer
        68 views

        Bringing a Heun equation into canonical form

        It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
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        3answers
        348 views

        (Sharp) inequality for Beta function

        I am trying to prove the following inequality concerning the Beta Function: $$ \alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0, $$ where as usual $B(a,b) = \...

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