Results 221 to 230 of about 48,005 (265)

Some explicit continued fraction expansions

Mathematika, 1991
The authors discuss the continued fraction expansion of infinite products \(\prod^{\infty}_{h=0}(1+x^{-\lambda_ h})\) which are elements of a field \(K((X^{-1}))\) where \(K\) is any field. The continued fraction is completely explicit provided the \(\lambda_ h\) are increasing integers such that \(\lambda_{h+1}/2\lambda_ h\) all are integers larger ...
Mendès France, Michel, van der Poorten, Alfred J.   +1 more
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Continued fraction inversion and expansion

IEEE Transactions on Automatic Control, 1979
The algorithm proposed by Parthasarthy and Singh for inverting continued fraction expansion in the Cauer first form is modified. Its use is extended for inverting a number of continued fraction expansions commonly encountered in engineering applications. An algorithm for converting a rational function into continued fraction is also given.
Rathore, T. S., Singhi, B. M., Kibe, A. V.   +2 more
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Oppenheim continued fraction expansion and Beta-expansion

Journal of Mathematical Analysis and Applications
For \(\beta>1\) and \(x\in[0,1)\), define a map \(T_\beta:[0,1)\to[0,1)\) as \(T_\beta(x)=\beta x-\left\lfloor\beta x\right\rfloor\), where \(\left\lfloor\cdot\right\rfloor\) denotes the integer part. Then, the \(\beta\)-expansion of \(x\) is given by \(x=\sum_{n=1}^\infty\varepsilon_n(x)/\beta^n\), where \(\varepsilon_n(x)=\left\lfloor\beta T_\beta^{n-
Yan Feng, Yuan Zhang
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Real Continued Fractions and Asymptotic Expansions

SIAM Journal on Mathematical Analysis, 1986
Let \(K(a_ n(x)/b_ n(x))\) be a continued fraction, where \(a_ n(x)\) and \(b_ n(x)\) are polynomials with nonnegative coefficients, in a real variable x. Let the continued fraction correspond at \(x=0\) to a formal power series in x or at \(x=\infty\) to a formal power series in \(x^{- 1}\).
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Continued Fraction Expansions

2010
A continued fraction [2, 3, 4, 5] is a sequence of fractions $$ f_n = b_0 + \frac{{a_1 }}{{b_1 + \frac{{a_1 }}{{b_2 + \frac{{a_3 }}{{b_3 + \cdots + \frac{{a_n }}{{b_n }}}}}}}} $$ (3.1) formed from two sequences a 1, a 2,… and b 0, b 1,… of numbers.
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On the Expansion of Ramanujan's Continued Fraction

The Ramanujan Journal, 1998
Ramanujan's continued fraction \(R(q)= 1+ \frac{q}{1+ \frac{q^2}{1+ \frac{q^3}{1+\dots}}}\) and its reciprocal \(R(q)^{-1}= \frac{1}{1+ \frac{q}{1+ \frac{q^2}{1+ \frac{q^3}{1+\dots}}}}\) can be expanded as series \[ \begin{aligned} R(q)&= 1+q-q^3+ q^5+\dots= \sum_{n=0}^\infty c(n)q^n \\ \text{and} R(q)^{-1}&= 1-q+q^2-q^4+\dots= \sum_{n=0}^\infty d(n) q^
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A New Z Domain Continued Fraction Expansion

IEEE Transactions on Circuits and Systems, 1982
A new Z domain continued fraction expansion is presented which proceeds in terms of z - 1 and 1 - z^{-1} factors. It is proved to be always convergent for polynomials whose roots all lie within the unit circle. The procedure involves a unique decomposition of the given polynomial into a mirror image polynomial (MIP) and an antimirror image polynomial ...
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Continued Fraction Expansions and the Legendre Polynomials

Bulletin of the London Mathematical Society, 1986
\textit{F. Beukers} has introduced a sequence of rational approximations to numbers of the form \(e^ a\), where a is rational, which arise from the Legendre expansion of \(e^{at}\), \(t\in [0,1]\) [Bull. Aust. Math. Soc. 22, 431-438 (1980; Zbl 0436.10016)].
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Several Continued Fraction Expansions of Generalized Cauchy Numbers

Bulletin of the Malaysian Mathematical Sciences Society, 2021
The author studies generalizations of the ordinary Cauchy numbers of the first kind \(c_n\) \[\frac{x}{\log{(1+x)}}=\sum_{n=0}^{\infty}\,c_n\,\frac{x^n}{n!}\tag{*}\] and several known continued fractions of the exponential generating function given above.
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Simultaneous Shifted Continued Fraction Expansions in Quadratic Time

Applicable Algebra in Engineering, Communication and Computing, 1998
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Niederreiter, Harald, Vielhaber, Michael
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