We use the elementary and analytic methods and the properties of Chebyshev polynomials to study the computational problem of the reciprocal sums of one‐kind Chebyshev polynomials and give several interesting identities for them. At the same time, we also give a general computational method for this kind of reciprocal sums.

In this paper, we give some interesting identities and asymptotic formulas for one kind of counting function, by studying the computational problems involving the symmetry sums of one kind quadratic residues and quadratic non-residues $\mathrm{mod} p$ . The main methods we used are the properties of the Legendre’s symbol $\mathrm{mod} p$ , and the estimate for character sums. As application, we solve two open problems proposed by Zhiwei Sun.

The ECL behaviors of NaBiF₄ : Yb³⁺/Er³⁺ UCNPs synthesized via a fast and environment-friendly method are reported for the first time. UCNPs-based ECL biosensor shows a wide detection range with low detection limit of 138 CFU mL⁻¹ for E. coli O157 : H7.

The main purpose of this paper is, using the elementary methods and properties of the power series, to study the computational problem of the convolution sums of Chebyshev polynomials and Fibonacci polynomials and to give some new and interesting identities for them.

The main purpose of this paper is using the combinatorial method, the properties of the power series and characteristic roots to study the computational problem of the symmetric sums of a certain second-order linear recurrence sequences, and obtain some new and interesting identities. These results not only improve on some of the existing results, but are also simpler and more beautiful. Of course, these identities profoundly reveal the regularity of the second-order linear recursive sequence, which can greatly facilitate the calculation of the symmetric sums of the sequences in practice.