[Paper Review] Fleck quotients and Bernoulli numbers

Zhi‐Wei Sun|ArXiv.org|Aug 14, 2006

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper extends Fleck's congruence and Wolstenholme's theorem by introducing higher-order Fleck quotients $ F_p^{(l)}(n,r) $, proving their congruence to products of binomial coefficients and higher-order Bernoulli numbers modulo $ p $. The key result establishes a $ p $-adic congruence linking these quotients to $ m $th-order Bernoulli numbers via $ F_p^{(l)}(n,r) ot o inom{loor{(n-l-1)/(p-1)}}{l} (n-l)_* B_{(n-l)^*}^{(m)}(-r) mod p $, generalizing classical binomial congruences to higher-order structures.

ABSTRACT

Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\sum_{k=r(mod p)}\binom{n}{k}(-1)^k\in\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ modulo p in terms of Bernoulli numbers, where m>0 is an integer with $m ot=n$ and $m=n (mod p(p-1))$. Consequently, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $00$ and $l\ge 0$ are integers with $2\le n-l\le p$ then $$\frac{1}{p^{n-l}}\sum_{l

Motivation & Objective

• To generalize Fleck's classical congruence $ inom{2p-1}{p-1} ot o 1 mod p^3 $ to higher-order $ p $-adic structures.
• To define and study higher-order Fleck quotients $ F_p^{(l)}(n,r) $, extending the original Fleck quotient $ F_p(n,r) $.
• To establish a $ p $-adic congruence linking $ F_p^{(l)}(n,r) $ to higher-order Bernoulli numbers $ B_k^{(m)}(-r) $.
• To unify and extend known results on binomial sums modulo $ p $, including Wolstenholme and Babbage's congruences.
• To provide a framework for understanding the $ p $-adic valuation of sums of signed binomial coefficients via higher-order Bernoulli polynomials.

Proposed method

• Introduces the higher-order Fleck quotient $ F_p^{(l)}(n,r) $ as a $ p $-adic refinement of the sum $ inom{n}{k}(-1)^k $ over $ k ot o r mod p $, scaled by $ (-p)^{-loor{(n-1)/(p-1)}} $.
• Applies the generating function for $ m $th-order Bernoulli polynomials $ B_k^{(m)}(t) $, defined via $ x^m e^{tx}/(e^x - 1)^m $, to express congruences.
• Uses Lucas' theorem and $ p $-adic expansions of integers to reduce $ F_p^{(l)}(p^{a-1}n + s, p^{a-1}r + t) $ modulo $ p $ to $ F_p^{(l)}(n,r) $.
• Employs induction and recursive decomposition of $ F_p^{(l)}(n,r) $ using the identity $ F_p^{(l)}(n,r) ot o inom{loor{(n-l-1)/(p-1)}}{l} F_p(n-lp, r) mod p $.
• Applies known results from [SW2] on $ F_p(n,r) ot o -n_*! B_{n^*}^{(m)}(-r) mod p $, with $ m ot o -n mod (p-1) $, to derive higher-order analogs.
• Uses the $ p $-adic order $ ext{ord}_p $ and properties of $ p $-adic integers $ bZ_p $ to ensure the Bernoulli numbers involved are $ p $-adically integral.

Experimental results

Research questions

• RQ1 How can Fleck’s classical congruence $ inom{2p-1}{p-1} ot o 1 mod p^3 $ be generalized to higher-order $ p $-adic structures?
• RQ2 What is the $ p $-adic behavior of the $ l $-th derivative of the Fleck quotient $ F_p^{(l)}(n,r) $, and how does it relate to Bernoulli numbers?
• RQ3 Can the sum $ inom{n}{k}(-1)^k $ over $ k ot o r mod p $ be expressed in terms of higher-order Bernoulli numbers modulo $ p $?
• RQ4 How does the $ p $-adic valuation of $ F_p^{(l)}(n,r) $ behave under scaling $ n o p^{a-1}n + s $, and what is the role of digit-wise expansion in $ p $-adic base?
• RQ5 Under what conditions does $ F_p^{(l)}(n,r) ot o 0 mod p $, and how is this tied to the non-vanishing of $ B_k^{(m)}(-r) $?

Key findings

• The higher-order Fleck quotient satisfies $ F_p^{(l)}(n,r) ot o -inom{loor{(n-l-1)/(p-1)}}{l} (n-l)_* B_{(n-l)^*}^{(m)}(-r) mod p $, where $ m ot o -n mod (p-1) $.
• For $ n > lp $, $ F_p^{(l)}(n,r) ot o inom{loor{(n-l-1)/(p-1)}}{l} F_p(n-lp, r) mod p $, establishing a recursive structure.
• When $ a ot o 1 $, $ F_p^{(l)}(p^{a-1}n + s, p^{a-1}r + t) ot o (-1)^t inom{s}{t} F_p^{(l)}(n,r) mod p $, generalizing to higher $ p $-powers.
• The congruence holds under conditions (i), (ii), or (iii) on the $ p $-adic digits of $ s $ and $ t $, with special treatment when $ s = t = p-1 $.
• The result extends Wolstenholme’s congruence: for $ 1 < n < p-1 $, $ rac{1}{p^n} inom{pn-1}{pk-1}(-1)^k ot o 0 mod p $, since $ B_{p-n} = 0 $.
• The $ p $-adic order of $ n! S(m heta(p^b), n) $ is $ loor{(n-1)/(p-1)} + ext{ord}_p(F_p(n,0)) $, linking Stirling numbers to Fleck quotients.