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Articles 31 - 34 of 34
Full-Text Articles in Number Theory
Real Numbers With Polynomial Continued Fraction Expansions, James Mclaughlin, Nancy Wyshinski
Real Numbers With Polynomial Continued Fraction Expansions, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
In this paper we show how to apply various techniques and theorems (including Pincherle’s theorem, an extension of Euler’s formula equating infinite series and continued fractions, an extension of the corresponding transformation that equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer-Muir transformation) to derive infinite families of in-equivalent polynomial continued fractions in which each continued fraction has the same limit. This allows us, for example, to construct infinite families of polynomial continued fractions for famous constants like π and e, ζ(k) (for each positive integer k ≥ 2), various special functions evaluated at …
Powers Of A Matrix And Combinatorial Identities, James Mclaughlin, B. Sury
Powers Of A Matrix And Combinatorial Identities, James Mclaughlin, B. Sury
Mathematics Faculty Publications
In this article we obtain a general polynomial identity in k variables, where k ≥ 2 is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a k × k matrix. Finally, we use these results to derive various combinatorial identities.
Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin
Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin
Mathematics Faculty Publications
In this paper we give a new formula for the n-th power of a 2 × 2 matrix. More precisely, we prove the following: Let A = (a b c d) be an arbitrary 2 × 2 matrix, T = a + d its trace, D = ad − bc its determinant and define yn : = b X n/2c i=0 (n − i i )T n−2i (−D) i . Then, for n ≥ 1, A n = (yn − d yn−1 b yn−1 c yn−1 yn − a yn−1) . We use this formula together with an existing formula …
Polynomial Continued Fractions, Douglas Bowman, James Mclaughlin
Polynomial Continued Fractions, Douglas Bowman, James Mclaughlin
Mathematics Faculty Publications
Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than or equal to one. Here we study cases of higher degree for both numerator and denominator polynomials, with particular attention given to cases in which the degrees are equal. We extend work of Ramanujan on continued fractions with rational limits and also consider cases where the limits are irrational.