**1**

vote

**2**answers

114 views

### 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 ...

**4**

votes

**1**answer

74 views

### Logarithm of the hypergeometric function

For $F(x)={}_2F_1 (a,b;c;x)$, with $c=a+b$, $a>0$, $b>0$, it has been proved in [1] that $\log F(x)$ is convex on $(0,1)$.
I numerically checked that with a variety of $a,\ b$ values, $\log ...

**4**

votes

**1**answer

149 views

### Conjectured bound on Kummer's function (confluent hypergeometric function)

I believe the following bound to be correct:
${}_1F_1(-a;1+a;-z) a z^{-a} \gamma(a,z) \leq 1$
for real-valued a > 0 and z $\geq 0$. $\gamma(a,z)$ is the lower incomplete gamma function.
Apart from ...

**26**

votes

**0**answers

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} ...

**6**

votes

**0**answers

295 views

### 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 ...

**5**

votes

**0**answers

377 views

### 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 ...

**4**

votes

**0**answers

292 views

### hypergeometric function $_2F_1(-n;-r;1;2)$

The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e.,
$$_2F_1(-n;-r;1;2) ...

**3**

votes

**0**answers

79 views

### Is there a way to express hypergeometric identities in terms of D-modules?

A hypergeometric function is a solution to
(*) w'' + p(z) w' + q(z) w = 0
where q has at most simple poles and q has at most double poles at 0,1,infty.
That differential equation is equivalent ...

**3**

votes

**0**answers

413 views

### Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?

I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...

**3**

votes

**0**answers

91 views

### Proving non-negativity of a hypergeometric type sum

I am trying to prove the following inequality:
$$
\sum\limits_{k=0}^{m}\frac{(a)_k(a+\mu)_{m-k}}{(c)_k(c+\mu)_{m-k}}\binom{m}{k}(m-2k+\mu)\geq{0},
$$
where $(a)_0=1$, $(a)_k=a(a+1)\cdots(a+k-1)$ is ...

**2**

votes

**0**answers

220 views

### Upper bounds on hypergeometric function 3F2

Are there any existing bounds on the hypergeometric function 3F2(-m,-n,1/2;1,1;4) where m,n>=1? Or any thoughts how one can be obtained?
Most of the inequalities I've seen apply to the cases whose ...

**1**

vote

**0**answers

66 views

### 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 ...

**1**

vote

**0**answers

175 views

### solving a sum of Hypergeometric function 2F3

I am trying to find a closed form solution for the two sums given by
$$\sum^n_{k=0}\frac{y^k}{k!}(-a+n)_k \left(\frac{2}{z} \right)^k {}_2F_3\left({-\frac{k}{2}, \frac{1 - k}{2}}; {-k, -a+n,1+a-k-n}; ...

**1**

vote

**0**answers

487 views

### Limit of two hypergeometric functions (2F1)

Hi,
Does anyone know whether there is a known function/distribution that corresponds to the limit:
$\lim_{\epsilon\rightarrow0^+} \mathfrak{Re}\left[f(x+i\epsilon) - f(x-i\epsilon)\right]$
when ...

**0**

votes

**0**answers

99 views

### On the Brioschi-like quintic $v^5 - 5d v^3 + 10 d^2 v - d^2 =0 $

The general quintic can be transformed in radicals using a rational Tschirnhausen transformation to the one-parameter Brioschi quintic,
$$u^5 - 10c u^3 + 45 c^2 u - c^2 = 0\tag{1}$$
which can be ...

**0**

votes

**0**answers

77 views

### How to simplify this 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 ...

**0**

votes

**0**answers

63 views

### When do the even part and odd part of a hypergeometric like function have common nonnegative real root?

I encountered the following problem in my research project:
Let
$$f(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$
We can separate the even part $p(x^2)$ from the odd part $x q(x^2)$ and write
...

**0**

votes

**0**answers

72 views

### Sum over Hypergeometric function 1F2

I would be very grateful for any ideas to find a closed form for the sum:
$$
\sum^\infty_{k=0}
\frac{z^k}{\Gamma(1+k) \Gamma(k+m+1)}
{}_1F_2\left(1;1+k,m+k;z\right)
$$
where $m\in\mathbb{N}$ and ...

**0**

votes

**0**answers

66 views

### A ${}_2 F_1$ equivalent of the Tricomi $U$ function?

The confluent hypergeometric function ${}_1F_1(a;b;z)$ has a natural partner in the Tricomi function $U(a,b,z)$, which provides a second, linearly independent solution to the confluent hypergeometric ...

**0**

votes

**0**answers

174 views

### Convert 2F1 to polynomial

Is there any transformation to convert each of the following versions of ${}_2F_1$ to a polynomial?
The first one is
$${}_2F_1\left(\frac{1-a}{2}, -\frac{a}{2}; b;\frac{4z}{(1+z)^2} \right), \quad ...