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Combinatorial congruences modulo prime powers. (English) Zbl 1119.11016

This very interesting paper contains several theorems concerning lower bounds on the \(p\)-adic order of sums of alternating binomial coefficients with special weights when the summation index is restricted to a congruence class modulo a power of the prime \(p\). For example, if \(\alpha\) and \(n\) are nonnegative integers, \(r\) is an integer, and \(f(X)\) is a polynomial with integer coefficients, then
\[ p^{\deg(f)} \sum_{k\equiv r\pmod{p^\alpha}} \binom nk (-1)^kf \biggl(\frac{k-r} {p^\alpha}\biggr)\equiv 0\pmod{p^{\sum_{i=\alpha}^\infty\lfloor n/p^{i}\rfloor}}. \] This is an extension of results of Fleck, C. S. Weisman and others. This congruence leads to a strong lower bound for the homotopy exponents of the special unitary group \(\text{SU}(n)\), as shown in a previous paper by the same two authors. An interesting corollary involving congruences for Bernoulli polynomials is presented. Another interesting result is the congruence
\[ \begin{aligned}\frac {1}{\lfloor n/p^{\alpha-1}\rfloor!}&\;\sum_{k\equiv r\pmod {p^\alpha}} \binom{pn+s}{pk+t} (-1)^{pk} \biggl(\frac{k-r}{p^{\alpha-1}}\biggr)^l\\ \equiv \frac {1}{\lfloor n/p^{\alpha-1}\rfloor!}&\;\sum_{k\equiv r\pmod {p^\alpha}} \binom nk \binom st (-1)^k\biggl(\frac{k-r}{p^{\alpha-1}}\biggr)^l \pmod p, \end{aligned} \] where \(\alpha\geq 2\) and \(l,s,t\) are nonnegative integers with \(s,t<p\). This is an extension of Lucas’ congruence
\[ \binom {pn+s}{pk+t}\equiv \binom nk \binom st\pmod p, \]
which can be obtained by specializing \(l=0\) and \(\alpha\) large with respect to \(n\) and \(k\) (say, \(p^{\alpha-1}>\max\{n,k\}\)) in the authors’ result (here, \(0^0:=1\)).
The proofs are elementary in nature but intricate. They are done by induction and use several known identities and congruences involving binomial coefficients such as binomial inversion, the Vandermonde identity, Lucas’ and Jacobstahl’s congruences, and, of course, Kummer’s theorem.
The paper contains several conjectures which are likely to spur further research on such topic.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
11A07 Congruences; primitive roots; residue systems
11B68 Bernoulli and Euler numbers and polynomials
11S05 Polynomials
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References:

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