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On the Identity of von Szily: Original Derivation and a New Proof



The original derivation of a 19th century identity associated with K. von Szily is presented and discussed. An independent proof is given using a technique developed a decade or so later (by J. Dougall) in relation to hypergeometric function theory. For completeness, a historical backdrop is provided for the reader, together with other relevant information.
... The von Szily identity [7,2,3] is ...
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The reciprocal Pascal matrix has entries $\binom{i+j}{j}^{-1}$. Explicit formullae for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, $q$-analogues are also presented.
My guess is that, within fifty or hundred years (or it might be one hundred and fifty) computers will successfully compete with the human brain in doing mathematics, and that their mathematical style will be rather different from ours. Fairly long computational verifications (numerical or combinatorical) will not bother them at all, and this should lead not just to different sorts of proofs, but more importantly to different sorts of theorems being proved.
The Catalan numbers are are well-known integers that arise in many combinatorial problems. The numbers , , and more generally are also integers for all n. We study the properties of these numbers and of some analogous numbers that generalize the ballot numbers, which we call super ballot numbers.