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Search: MSC category 11B37 ( Recurrences {For applications to special functions, see 33-XX} )

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1. CMB 2011 (vol 56 pp. 283)

Coons, Michael
 Transcendental Solutions of a Class of Minimal Functional Equations We prove a result concerning power series $f(z)\in\mathbb{C}[\mkern-3mu[z]\mkern-3mu]$ satisfying a functional equation of the form $$f(z^d)=\sum_{k=1}^n \frac{A_k(z)}{B_k(z)}f(z)^k,$$ where $A_k(z),B_k(z)\in \mathbb{C}[z]$. In particular, we show that if $f(z)$ satisfies a minimal functional equation of the above form with $n\geqslant 2$, then $f(z)$ is necessarily transcendental. Towards a more complete classification, the case $n=1$ is also considered. Keywords:transcendence, generating functions, Mahler-type functional equationCategories:11B37, 11B83, , 11J91

2. CMB 2011 (vol 55 pp. 60)

Coons, Michael
 Extension of Some Theorems of W. Schwarz In this paper, we prove that a non--zero power series $F(z)\in\mathbb{C} [\mkern-3mu[ z]\mkern-3mu]$ satisfying $$F(z^d)=F(z)+\frac{A(z)}{B(z)},$$ where $d\geq 2$, $A(z),B(z)\in\mathbb{C}[z]$ with $A(z)\neq 0$ and $\deg A(z),\deg B(z) Keywords:functional equations, transcendence, power seriesCategories:11B37, 11J81 3. CMB 2004 (vol 47 pp. 12) Burger, Edward B.  On Newton's Method and Rational Approximations to Quadratic Irrationals In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio\break$(-1+\sqrt5)/2$for which when Newton's method for approximating roots is applied with an initial value$x_0=0$, all approximates are so-called best rational approximates''---in this case, of the form$F_{2n}/F_{2n+1}$, where$F_n$denotes the$n$-th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length$2\$. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena. Categories:11A55, 11B37

4. CMB 2001 (vol 44 pp. 19)

Brindza, B.; Pintér, Á.; Schmidt, W. M.
 Multiplicities of Binary Recurrences In this note the multiplicities of binary recurrences over algebraic number fields are investigated under some natural assumptions. Categories:11B37, 11J86