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Sorry, my answer is wrong. As it was pointed out by "Simply Beautiful Art" $x\ne W(z)$ but $x=e^{W(z)}.$ It is known that $$W'(z)=\frac{W(z)}{z(1+W(z))}.$$ Let $z=\log t$. Then $x=W(z)$ is a root of …

There are fifteen Gauss' contiguous relations for functions $F(a, b ; c ; z)$, $F(a \pm 1, b ; c ; z), F(a, b \pm 1, c ; z)$, and $F(a, b ; c \pm 1 ; z)$, see Erdélyi A. et al. Higher transcendental f …

There is one natural way to study such identities based on discrete Fourier series. You can define functions as a Fourier series with coefficients equal to your summands. Then your sums will be equal …

You can find this expansion in the book Lorentzen L. & Waadeland H. Continued fractions with applications North-Holland Publishing Co., 1992 (formula (4.3.10)). As I understand, this document is a mor …

In the question Eigenvalues of a matrix with entries involving combinatorics No_way asked about eigenvectors of $n\times n$ matrix $M$ with entries \begin{eqnarray*} M_{ij}=(-1)^{i+j}F(n, l, i, j), \ …

I'm trying to read the article "Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups" by Bukhshtaber,V. M. Russian Mathematical Surveys(1990),4 …

There is a more simple proof in the book V. M. Buchstaber, T. E. Panov, Toric Topology, Mathematical Surveys and Monographs, 204, Amer. Math. Soc., 2015, arXiv: 1210.2368v3. This result is the theorem …

Let $P_n(x)$ be Legendre polynomials: $$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$ Usual arguments from the theory of formal groups allow to prove that for any $n$ $$P_n(x)=Q_n( …

These numbers and corresponding polynomials were introduced by Korobov, see second edition of his book "Number theory methods in numerical analysis". He found some examples, where these "discrete" pol …

One can find a good collection of summation formulae in Interpolation by J. F. Steffensen (1950). §12. Laplace’s and Gauss’s Summation-Formulas §14. Euler’s Summation-Formula §15. Lubbock’s and Woolh …