Q-Congruences And Lacunary Sums Of Trinomial Coefficients

q-Congruences and sums of trinomial coefficients are important parts of Combina-torial Number Theory.In this thesis we mainly study some symmetric q-congruences.q-congruences involving the Jacobi symbol,the divisibility of some binomial sums,and explicit formulas for lacunary sums of trinomial coefficients.q-Congruences are related to many fields of mathematics,such as partition theory,hypergeometric series,and the CFT in biophysics.We adopt the following standard notation: In this thesis,by using tools of combinatorial identities and cyclotomic polynomials,we mainly prove the following few q-congruences:(i)Let n>1 be an integer and let d>0 and r be integers with(n,d)=1. Suppose that f1(q),…,fn(g)∈Z[g]and If n is odd,then when n is even,we have where a denotes the least nonnegative integer x with x≡-r/d(mod n),and let φn(q)be the n-th cyclotomic polynomial in q.This implies a conjecture of Victor J.W.Guo and J.Zeng,and a congruence of Z.-H.Sun.(ii)Let n,d>1 be two integers with(d,n)=1.Let r∈Z and let a be the least non-negative residue of-r d modulo n.Then,for any s=0,1,...,n-a-1 we have This implies a conjecture of V.J.W.Guo and J.Zeng.(iii)Let m be a positive integer relatively prime to a positive odd integer n.Then Where(m/n)denotes the Jacobi symbol.This confirms a conjecture of V.J.W.Guo on a q-analogue of Euler’s congruence mp-1/2(m/p)(mod p)with p an odd prime.(iv)We prove the following conjecture of Guo 11,Conjecture 5.4]:for any integers and Actually,we obtain a more general result.For a positive integer n,the trinomial coefficients(nk)2(k∈Z)are given by For integers m>0 and r,we express the lacunary sum∑k≡r(mod m)(nk)2 are the sum of some linear recurrences with integer coefficients.This extends Andrews’ formulas on ∑k≡r(mod 10)(nk)2.This thesis consists of six chapters.In the first chapter,we present a survey of q-congruences and introduce known extensions of lacunary sums of trinomial coefficients,and also state our main results.Chapters 2-6 are devoted to our proofs of the main results in this thesis.Parts(ii)and(iii)and our results on lacunary sums of trinomial coefficients have been published or accepted for publication,and our papers containing parts(i)and(iv)are publicly available from arXivorg.