Tagged Questions

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Goin' with the flow with Kummer and Pascal: Combinatorics and geometry underlying the logarithm of the derivative operator

In a MO-Q111165 and associated MSE-Q125343, I present a pair of raising / lowering (creation / annihilation) operators $R_x = log(D)$ and $L_x = -x·D$ with $D=d/dx$ (for a sequence of functions ...
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Under which constraints are there only finite numbers of irreducible eta product identities?

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 $e_k:=\eta(q^k)$. An eta product identity (or eta identity for ...
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A definite integral of hypergeometric function 2F1

I am wondering whether there exists a closed form for the definite integral $$F(x)=\int_0^1t^{-a}(1-t)^{N}(1-xt)^{-a}{}_2F_1(-a,k-a-1/2,k-a;4xt(1-xt))dt,$$ where $a\in(0,1)$ and $N,k$ are positive ...
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Quadratic transformation of hypergeometric function 2F1

I want to know whether there is some transformation between $_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$. Here is an example called the Kummer quadratic transformation, which may be known to most of ...
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For integers $n \geq k \geq 0$, can anyone provide a proof for the following identity? $$\sum_{j=0}^k\left(\begin{array}{c}2n+1\\\ 2j\end{array}\right)\left(\begin{array}{c}n-j\\\ ... 1answer 363 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 ... 1answer 345 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 ... 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 ... 0answers 279 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 ... 2answers 589 views Cosine of a Partial Sum Does anyone know of a closed formula for cos(\displaystyle\sum_{n=0}^m a_{n})? I've seen formulas for cos(\displaystyle\sum_{n=0}^\infty a_{n}) and tan(\displaystyle\sum_{n=0}^m a_{n}), but the ... 2answers 1k views Product of hypergeometric functions/Jacobi Polynomials Are there any theorems related to the product of Jacobi/Legendre Polynomials and/or Hypergeometric functions? Specifically, I'm interested in the product of {}\_{2}F_{1}[-n,-n+1;2;x] and ... 4answers 2k views Groups, quantum groups and (fill in the blank) In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ... 1answer 769 views Is this sequence of polynomials well-known? While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define ... 0answers 1k views Curious q-analogues Consider the Fibonacci polynomials$$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$and the Lucas polynomials$$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ...
For integers $n\ge2$ and $k\ge2$, fix the notation $$[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad [m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.$$ Consider the following trigonometric ...