In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extende...In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory.展开更多
In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expre...In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.展开更多
基金supported in part by the National Natural Science Foundation of China(Grant No.12061033)by the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region(Grants No.NJZY20119)by the Natural Science Foundation of Inner Mongolia(Grant No.2019MS01007),China.
文摘In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory.
基金The first author,Mrs.Yan Hong,was partially supported by the Natural Science Foundation of Inner Mongolia(Grant No.2019MS01007)by the Science Research Fund of Inner Mongolia University for Nationalities(Grant No.NMDBY15019)by the Foun-dation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region(Grant Nos.NJZY19157 and NJZY20119)in China。
文摘In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.
