A simple proof of Ramanujan’s summation of the 1 ω 1. Aequationes Math, 1978, 18: 333–337 MATH MathSciNetG. E. Andrews and R. Askey, A simple proof of Ramanujan's summation of the 1ψ1, Aequationes Math. 18 (1978), 333-337....
3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . ....
3 Ramanujan’s Proof of the q-Gauss Summation Theorem . . . . . 10 1. 4 Corollaries of (1. 2. 1) and (1. 2. 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1. 5 Corollaries of (1. 2. 6) and (1. 2. 7) . . . . . . . . ....
H. Hardy described what we now call Ramanujan's famous \\(_1\\psi_1\\) summation theorem as "a remarkable formula with many parameters." This is now one of the fundamental theorems of the subject.Despite humble beginnings, the ... ...
By means of the extended Gould-Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well--poised $_7F_6$-series can be utilized to construct numerous interesting Ramanujan--like infinite series expressions for $pi^{pm1}$ and $pi^{pm2}$, inc...
Ramanujan's Summation FormulaQuintuple Product IdentityThis unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details...
0 n,m The resulting bracket series has more summation indices than brackets. The choice of n as a free variable, gives m∗ = −n − s and Rule 2 produces the convergent series (9.9) ∞ (−1)n n! Γ(n + s) = Γ(s)1F0 s − −1 Γ(s) = 2s . n=0 Symmetry ...
Proof From (3.26), we clearly see that A 2 ( 2 n ) > 0 for all n > 2 . Therefore, ( − 1 ) l F 2 , n ( l ) ( a ) ≥ 0 for all a ∈ ( 0 , 1 / 2 ] and l = 0 , 1 , 2 , 3 … , and F 2 , n is completely monotonic on ( 0 , 1 / 2 ] .□ ...
we provide an analytic proof of the result of Ariki and Mathas for the above choice of parameters. Our second objective is to investigate simple modules of the Ariki–Koike algebra in a fixed block, which are, as widely known, labeled by the Kleshchev multiparitions with a fixed partition...
(1/2,g⊗χ)is of Burgess strength. The bound was proved by a couple of methods: shifted convolution sums and the Petersson/Kuznetsov formula analysis. It is natural to ask what inputs are really needed to prove a Burgess-type bound onGL(2). In this paper, we give a new proof of ...