Result: Old and new formulas for the Hopf–Stiefel and related functions: Old and new formulas for the Hopf-Stiefel and related functions
Further Information
The Hopf-Stiefel function is defined on pairs \(a,b\in\mathbb{N}\) by \[ a\circ b= \min\biggl\{n\in\mathbb{N}\;\biggl|\;{n\choose k}\equiv 0\pmod 2\text{ for all integers }k\in \{n-a< k< b\}\biggr\} \] [\textit{H. Hopf}, Comment. Math. Helv. 13, 219--239 (1941; Zbl 0024.36002); \textit{E. Stiefel}, Comment. Math. Helv. 13, 201--218 (1941; Zbl 0024.36001; JFM 67.0737.02)]. In this paper first some equivalent formulas for \(a\circ b\) are listed, for instance the recursion formulas \(a\circ 1= a\), \(a\circ b= b\circ a\), \(a\circ b= 2^k\) if \(2^{k-1}< a\), \(b\leq 2^k\) and \(a\circ b= 2^k+ a\circ (b- 2^k)\) if \(2^{k-1}< a\leq 2^k< b\) due to \textit{A. Pfister} [J. Lond. Math. Soc. 40, 159--165 (1965; Zbl 0131.25002)] who used these formulas as the definition and applied the representations by quadratic forms. Further some generalizations of the Hopf-Stiefel function are given and relations between them are proved. As a related function to the Hopf-Stiefel function the function \(K\) is also discussed defined for a sequence \[ {\mathcal H}= \{h_0= 1< h_1< h_2