The main aim of this paper is that for any second-order linear recurrence sequence, the generating function of which is f ( t ) = 1 1 + a t + b t 2 , we can give the exact coefficient expression of the power series expansion of f x ( t ) for x ∈ R with elementary methods and symmetry properties. On the other hand, if we take some special values for a and b, not only can we obtain the convolution formula of some important polynomials, but also we can establish the relationship between polynomials and themselves. For example, we can find relationship between the Chebyshev polynomials and Legendre polynomials.
Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ±3(mod 8), then the equation 8x + p
y = z
2 has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod 8), then the equation has only the solutions (p, x, y, z) = (2q − 1, (1/3)(q + 2), 2, 2q + 1), where q is an odd prime with q ≡ 1(mod 3); (iii) if p ≡ 1(mod 8) and p ≠ 17, then the equation has at most two positive integer solutions (x, y, z).
In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.
Let n be a positive integer and k≥2, bk(n) denotes the k-th power complement fuction, we define a new set A.This paper is mainly to study the mean value properties of the Euler function in set A,and give an interesting asymptotic formula.
In this paper, we use the analysis method and the properties of trigonometric sums to study the computational problem of one kind power mean of the hybrid Gauss sums. After establishing some relevant lemmas, we give an exact computational formula for it. As an application of our result, we give an exact formula for the number of solutions of one kind diagonal congruence equation mod p, where p be an odd prime.
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