Abstract: By introducing several positive parameters, a new fractional discrete kernel function is constructed. By means of the method of weight coefficients, a double Hilbert series inequality is established, and the constant factor of the newly obtained inequality is proved to be the best possible. Furthermore, based on the rational fraction expansion of the cosecant function, it is also proved that the optimal constant factor can be represented by the cosecant function. At last, by assigning some specific values to the parameters, some previously known results are obtained, and some new Hilbert-type inequalities with special kernel functions are also presented at the end of the paper.

Key words: Hilbert-type inequality, fractional kernel function, rational fraction expansion, cosecant function