All Questions
Tagged with determinants special-functions
7 questions
7
votes
5
answers
1k
views
How to calculate one Cauchy type determinant
As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y _i)}{\...
- 113
6
votes
1
answer
535
views
A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$
On the basis of my computation, here I pose my following conjecture involving the cosine function.
Conjecture. For any positive integer $n$, we have the identity
$$\frac1{2n}\det\left[\cos\pi\frac{jk}...
- 15.6k
6
votes
0
answers
266
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
$$S_p:=\det\left[\sec2\pi\frac{jk}p\right]_{0\le j,k\le (...
- 15.6k
4
votes
0
answers
96
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 43.2k
1
vote
1
answer
220
views
A determinant involving the cotangent function
Let $n>1$ be odd. In my 2019 preprint On some determinants involving the tangent function, I proved that
$$\det\left[\tan\pi\frac{aj+bk}n\right]_{1\le j,k\le n-1}=\left(\frac{-ab}n\right)n^{n-2}$$
...
- 15.6k
0
votes
1
answer
188
views
Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials
Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials)
\begin{equation}
f(1,\frac{1}{2},...
- 11
0
votes
0
answers
164
views
How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?
The classical Euler gamma function can be defined by the integral
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0.
\end{equation*}
Its ...
- 1,091