In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presen...In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presented, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions are derived, and two double inequalities for bounding the ratio of two gamma functions are discovered.展开更多
Here is introduced some novel algorithms which made use of polygarnma functions to get the exact limits of a broad class of infinite series. Moreover, Laplace transform is used to find the sum of many convergent infin...Here is introduced some novel algorithms which made use of polygarnma functions to get the exact limits of a broad class of infinite series. Moreover, Laplace transform is used to find the sum of many convergent infinite series. These exact limits are found in different branches of physics for some special cases series and are in complete agreement with the values found by other authors. Moreover, the methods presented here are generalized and applied to other wide variety of sums, including alternating series. Finally, these methods are simple and quite powerful to calculate the limits of many convergent series as you can see from the examples included.展开更多
基金supported partially by the China Scholarship Council and the Science Foundation of Tianjin Polytechnic Universitysupported in part by the Natural Science Foundation Project of Chongqing,China(Grant No.CSTC2011JJA00024)+1 种基金the Research Project of Science and Technology of Chongqing Education Commission,China(Grant No.KJ120625)the Fund of Chongqing Normal University,China(Grant Nos.10XLR017 and 2011XLZ07)
文摘In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presented, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions are derived, and two double inequalities for bounding the ratio of two gamma functions are discovered.
文摘Here is introduced some novel algorithms which made use of polygarnma functions to get the exact limits of a broad class of infinite series. Moreover, Laplace transform is used to find the sum of many convergent infinite series. These exact limits are found in different branches of physics for some special cases series and are in complete agreement with the values found by other authors. Moreover, the methods presented here are generalized and applied to other wide variety of sums, including alternating series. Finally, these methods are simple and quite powerful to calculate the limits of many convergent series as you can see from the examples included.
