[Paper Review] Congruences involving generalized central trinomial coefficients

Zhi‐Wei Sun|arXiv (Cornell University)|Aug 23, 2010

Advanced Mathematical Identities6 references4 citations

TL;DR

This paper establishes new congruences involving generalized central trinomial coefficients $ T_n(b,c) $ and generalized Motzkin numbers $ M_n(b,c) $, proving that certain weighted sums modulo prime powers exhibit deep arithmetic structure tied to Legendre symbols and quadratic residues. The key contribution is a set of exact congruences modulo $ p $ and $ p^2 $, extending classical results on central trinomial and Motzkin numbers, with explicit formulas involving $ inom{2k}{k} $, harmonic numbers, and Legendre symbols.

ABSTRACT

For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\sum_{k=0}^n\binom nk\binom{n+k}k$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\ldots$ we have $$\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}\equiv0\pmod{n^2}$$ and in particular $n^2\mid\sum_{k=0}^{n-1}(2k+1)D_k^2$; if $p$ is an odd prime then $$\sum_{k=0}^{p-1}T_k^2\equiv\left(\frac{-1}p ight)\ \pmod{p}\ \ \ { m and}\ \ \ \sum_{k=0}^{p-1}D_k^2\equiv\left(\frac 2p ight)\ \pmod{p},$$ where $(-)$ denotes the Legendre symbol. We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

Motivation & Objective

• To investigate arithmetic properties of generalized central trinomial coefficients $ T_n(b,c) $ and generalized Motzkin numbers $ M_n(b,c) $, extending classical central trinomial and Motzkin numbers.
• To establish new congruences modulo $ p $ and $ p^2 $ for sums involving $ T_n(b,c) $, $ M_n(b,c) $, and their products, with weights involving $ d = b^2 - 4c $ and $ m $-powers.
• To generalize known results on central trinomial and Motzkin numbers by introducing parameters $ b, c $, and to explore connections to Legendre polynomials and Delannoy numbers.
• To formulate and support conjectures on higher-order congruences modulo $ p^3 $, $ p^4 $, and for sums with linear weights, based on patterns observed in $ p $-adic valuations.
• To unify and extend prior work on central trinomial and Motzkin number congruences by introducing a parameterized framework using generating functions and hypergeometric identities.

Proposed method

• Derive generating functions for $ T_n(b,c) $ and $ M_n(b,c) $, showing $ rac{1}{insqrt{1 - 2bx + dx^2}} $ for $ T_n(b,c) $ and a related algebraic expression for $ M_n(b,c) $, with $ d = b^2 - 4c $.
• Use the Zeilberger algorithm to derive recurrence relations for $ M_n(b,c) $, and use known recurrences for $ T_n(b,c) $ to analyze their $ p $-adic behavior.
• Apply the Legendre symbol $ ig(rac{a}{p}ig) $ to characterize congruences modulo $ p $, particularly in sums involving $ rac{T_k(b,c)}{m^k} $ and $ rac{M_k(b,c)}{m^k} $.
• Use the identity $ T_n(b,c) = ( ext{discriminant})^{n/2} P_n(b/ ext{disc}) $ to relate $ T_n(b,c) $ to Legendre polynomials $ P_n(x) $, enabling evaluation of sums modulo $ p $.
• Employ the theory of hypergeometric sums and $ p $-adic analysis to derive congruences modulo $ p^2 $, especially for $ rac{T_k^2}{d^k} $, $ rac{T_k M_k}{d^k} $, and weighted sums.
• Formulate conjectures based on numerical patterns and known identities, particularly for sums involving $ T_n(b,c)^3 $, $ T_n(b,c)^2 M_n(b,c) $, and linear weights $ (ak+b) $, with moduli $ p^2 $, $ p^3 $, and $ p^4 $.

Experimental results

Research questions

• RQ1 What are the congruence properties of $ rac{T_k(b,c)}{m^k} $ modulo $ p $, and how do they depend on the Legendre symbol $ ig(rac{(m-b)^2 - 4c}{p}ig) $?
• RQ2 How do the sums $ rac{M_k(b,c)}{m^k} $ behave modulo $ p $, and what role does the discriminant $ (m-b)^2 - 4c $ play in their structure?
• RQ3 What is the value of $ rac{T_k(b,c)^2}{d^k} mod p $, and how does it relate to the Legendre symbol $ ig(rac{cd}{p}ig) $?
• RQ4 Can the sum $ rac{T_k(b,c)M_k(b,c)}{d^k} mod p $ be shown to vanish under certain conditions on $ b, c $, and $ d $?
• RQ5 What higher-order congruences emerge for $ rac{T_k(b,c)^3}{(b-2c)^{3k}} mod p^2 $, and how do they depend on the splitting of primes in quadratic forms?

Key findings

• For any odd prime $ p $, $ rac{1}{m^k} $-weighted sum of $ T_k(b,c) $ modulo $ p $ satisfies $ extstyleig(rac{(m-b)^2 - 4c}{p}ig) $, linking it to quadratic residues.
• The $ m^{-k} $-weighted sum of $ M_k(b,c) $ modulo $ p $ is congruent to $ (m-b)^2 - ((m-b)^2 - 4c)ig(rac{(m-b)^2 - 4c}{p}ig) $, scaled by $ 2c $.
• The sum $ extstylerac{T_k(b,c)^2}{d^k} mod p $ is congruent to $ ig(rac{cd}{p}ig) $, provided $ d ot o 0 mod p $.
• For $ b ot o 2c mod p $, the sum $ extstylerac{T_k(b,c^2)^2}{(b-2c)^{2k}} mod p $ is congruent to $ ig(rac{-c^2}{p}ig) $.
• If $ p mid c $ and $ d ot o 0 mod p $, then $ extstylerac{T_k(b,c)M_k(b,c)}{d^k} mod p $ vanishes, indicating deep arithmetic cancellation.
• For $ D = b^2 - 4c^2 ot o 0 mod p $, the sum $ extstylerac{T_k(b,c^2)M_k(b,c^2)}{(b-2c)^{2k}} mod p $ is congruent to $ rac{4b}{b+2c} ig(rac{D}{p}ig) $.