An overpartition ofnis a non-increasing sequence of natural numbers whose sum isnin which the first occurrence of a number may be overlined and\overline{p}(n)denotes the number of overpartitions ofn. For convenience, define\overline{p}(0)=1. For example, there are 8 overpartitions of 3 ...
in the sense that his work will be useful and interesting for many years to come.” Hardy also said that Ramanujan’s genius was “a natural gift which heaven had bestowed upon him in a pre-eminent degree”.
This work is supported by Beijing Natural Science Foundation (Grant No. 1242003), and the National Natural Science Foundation of China (Grant Nos. 11901566, 12001047, 11971476, 12071238). Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive ri...
Through the study of the numbers of permutations and maxim which have the same reverse order , we get the recurring formula,with which we can extend the Kendall τ cofficient table which is frequently applied in mathematical statistics in the condition of any natural number. 由此将数理统计中常...
Thus it is very natural for Hin- dus to accept the Goddess of Namakkal as a divine origin of Ramanujan's great discoveries. In England, within a short span of five years, Ramanujan wrote several funda- mental papers, some with Hardy, that revolutionized various areas of math- ematics. ...
[22] proved a recurrence and a generating function formula for the, and it is this formula which eventually yields Eq. (1). One natural generalization would be to consider surjective functionswhich satisfyfor any finite. Given the role that the generalized Dumont–Foata polynomials play in [14...
Numbers and Mathematics The distinction between discrete mathematics and continuous mathematics is a central issue for mathematical modeling, the art of representing features of the natural world in mathematical form. The universe does not contain or consist of actual mathematical objects, but many aspect...
In the present context, these are composed of two finite component sets, of equal cardinality, of natural numbers, such that the numbers formed by considering all sums and all absolute differences of all pairs of numbers, each taken from one of the component sets, with or without inclusion ...
In this paper, we obtain analogues of Jacobi’s derivative formula in terms of the theta constants with rational characteristics. For this purpose, we use the arithmetic formulas of the number of representations of a natural number n [Math Processing Error](n=1,2,…) as the sum of two squ...
Supported by the National Natural Science Foundation (Grant No. 11571162) of China. Rights and permissions Reprints and permissions About this article Cite this article Sun, ZW. Congruences involving gn(x)=∑k=0n(nk)2(2kk)xk . Ramanujan J 40, 511–533 (2016). https://doi.org/10.1007/...