# Tagged Questions

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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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### Polynomials with presumably positive coefficients

The $q$-Pochhammer symbol $(q) _ 0:=1$ and $(q) _ n:=\prod _ {j=1}^n (1-q^j)$ for $n > 0$ is clearly a polynomial in $q$ which has both positive and negative coefficients when $n>0$. The ...
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### 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 ...
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### Relation between two hypergeometric series

EDIT: Context for this investigation can be found in one of my other MO posts, "Pythagorean Theorem for Right-Corner Hyperbolic Simplices?" -- I'm investigating a function that has led me to this ...
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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 $$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ... 1answer 715 views ### Is there a closed form for this hypergeometric expression? I am trying to compute the number of distinct ways a 4n \times 4n chessboard can be colored black and white, with exactly half the squares black and half the squares white. By distinct, I mean ... 1answer 915 views ### A (known?) hypergeometric identity Incidentally I've obtained a hypergeometric identity that I've not seen before:$${}_3F_2(-m,-n,m+n; 1, 1; 1) = \frac{m^2+n^2+mn}{(m+n)^2} {\binom{m+n}{m}}^2$$So, I wonder if it is well-known and ... 1answer 385 views ### “Closed” form for Motzkin and related numbers I wonder whether it is impossible to write the nth Motzkin number as a sum of a fixed number of, say, hypergeometric terms. To illustrate what I mean: n!+(2n)! is not a hypergeometric term, but it ... 4answers 2k views ### Integer-valued factorial ratios This historical question recalls Pafnuty Chebyshev's estimates for the prime distribution function. In his derivation Chebyshev used the factorial ratio sequence$$ ...
The generating function $f(z)$ of the Catalan numbers which is characterized by $f(z)=1+zf(z)^2$ is D-finite, or holonomic, i.e. it satisfies a linear differential equation with polynomial ...