Parametric Evaluations of the Rogers-Ramanujan Continued Fraction
نویسنده
چکیده
In this paper with the help of the inverse function of the singular moduli we evaluate the Rogers-Ranmanujan continued fraction and its first derivative. 1 q 1 1 q 2 1 q 3 1 · · ·. 1.1 We also define a; q n : n−1 k0 1 − aq k , f −q : ∞ n1 1 − q n q; q ∞ .
منابع مشابه
Two-dimensional Series Evaluations via the Elliptic Functions of Ramanujan and Jacobi
We evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers–Ramanujan continued fraction play central roles in our evaluations.
متن کاملOn the Generalized Rogers–ramanujan Continued Fraction
On page 26 in his lost notebook, Ramanujan states an asymptotic formula for the generalized Rogers–Ramanujan continued fraction. This formula is proved and made slightly more precise. A second primary goal is to prove another continued fraction representation for the Rogers–Ramanujan continued fraction conjectured by R. Blecksmith and J. Brillhart. Two further entries in the lost notebook are e...
متن کاملThe Rogers-Ramanujan continued fraction and its level 13 analogue
One of the properties of the Rogers-Ramanujan continued fraction is its representation as an infinite product given by r(q) = q ∞ ∏
متن کاملON THE DIVERGENCE IN THE GENERAL SENSE OF q-CONTINUED FRACTION ON THE UNIT CIRCLE
We show, for each q-continued fraction G(q) in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which G(q) diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other q-continue...
متن کاملOn the Divergence of the Rogers-ramanujan Continued Fraction on the Unit Circle
This paper is an intensive study of the convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, a1(t), a2(t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted by ci(t)/di(t). Let S = {t ∈ (0, 1) : ai+1(t) ≥ φi infinitely often}, where φ = ( √ 5+1)/2. Let YS = {exp(2πit) ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Int. J. Math. Mathematical Sciences
دوره 2011 شماره
صفحات -
تاریخ انتشار 2011