Jacobian Sequences And Polynomial Expansion Coefficients

Let p_k/q_k,k≥0, be the convergents of the continued fraction expansion of an irrational real number θ. The sequence of Jacobi symbols （pk/qk）,k≥0 is called the Jacobi sequence of θ. Girstmair showed that the Jacobi sequence of e is purely periodic with period length 24 and the Jacobi sequence of e² is purely periodic with period length 40. In this thesis, we determine the minimal lengths of positive periods of the Jacobi sequences for θ= （?） （n≥1） and （? ） （n≥2）.Let Mm（n,k） be the number of multinomial coefficients such that where k₁+k₂+…k_n= k. In 2014, Merca gave the formula for Mp（n,k） when p is prime. Recently, Guo confirmed a conjecture of Merca by extending Merca’s result to all prime powers m= p^r. In this thesis, we consider M_m（n,k） for all positive integers m. As a corollary of our results, we give the formula for M_p^αq^β（n,k）, where p, q are distinct primes and α, β are nonnegative integers with α+β>0.