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Articles 31 - 42 of 42
Full-Text Articles in Number Theory
Ramanujan And Extensions And Contractions Of Continued Fractions, James Mclaughlin, Nancy Wyshinski
Ramanujan And Extensions And Contractions Of Continued Fractions, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
If a continued fraction K∞n=1an/bn is known to converge but its limit is not easy to determine, it may be easier to use an extension of K∞n=1an/bn to find the limit. By an extension of K∞n=1an/bn we mean a continued fraction K∞n=1cn/dn whose odd or even part is K∞n=1an/bn. One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if K∞n=1an/bn is the even part); (ii) Find the limit of the …
Continued Fractions And Generalizations With Many Limits: A Survey, Douglas Bowman, James Mclaughlin
Continued Fractions And Generalizations With Many Limits: A Survey, Douglas Bowman, James Mclaughlin
Mathematics Faculty Publications
There are infinite processes (matrix products, continued fractions, (r, s)-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way. We give a survey of results in this area, focusing on recent results of the authors.
Further Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin, Nancy Wyshinski
Further Combinatorial Identities Deriving From The N-Th Power Of A 2 X 2 Matrix, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
In this paper we use a formula for the n-th power of a 2×2 matrix A (in terms of the entries in A) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if m and n are positive integers and s ∈ {0, 1, 2, . . . , b(mn − 1)/2c}, then X i,j,k,t 2 1+2t−mn+n (−1)nk+i(n+1) 1 + δ(m−1)/2, i+k m − 1 − i i ! m − 1 − 2i k ! × n(m − 1 − 2(i + k)) 2j ! j t − n(i + k) ! n …
A Q-Continued Fraction, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi
A Q-Continued Fraction, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi
Mathematics Faculty Publications
Let a, b, c, d be complex numbers with d 6= 0 and |q| < 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq2n+1 + cqn (a + b)q n+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d) j (−c/bd)j q j(j+3)/2 (q)j (−aq2/d)j P∞ j=0 (b/d) j (−c/bd)j q j(j+1)/2 (q)j (−aq/d)j . We then use this result to deduce various corollaries, including the following: 1 1 − q 1 + q − q 3 1 + q 2 − q 5 1 + q 3 − · · · − q 2n−1 1 + q n − · · · = (q 2 ; q 3 )∞ (q; q 3)∞ , (−aq)∞ X∞ j=0 (bq) j (−c/b)j q j(j−1)/2 (q)j (−aq)j = (−bq)∞ X∞ j=0 (aq) j (−c/a)j q j(j−1)/2 (q)j (−bq)j , and the Rogers-Ramanujan identities, X∞ n=0 q n 2 (q; q)n = 1 (q; q 5)∞(q 4; q 5)∞ , X∞ n=0 q n 2+n (q; q)n = 1 (q 2; q 5)∞(q 3; q 5)∞.
The Convergence Behavior Of Q-Continued Fractions On The Unit Circle, Douglas Bowman, James Mclaughlin
The Convergence Behavior Of Q-Continued Fractions On The Unit Circle, Douglas Bowman, James Mclaughlin
Mathematics Faculty Publications
In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of qcontinued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, YG, on the unit circle such that if y ∈ YG then G(y) does not converge to a finite value. We discuss …
The Convergence And Divergence Of Q-Continued Fractions Outside The Unit Circle, Douglas Bowman, James Mclaughlin
The Convergence And Divergence Of Q-Continued Fractions Outside The Unit Circle, Douglas Bowman, James Mclaughlin
Mathematics Faculty Publications
We consider two classes of q-continued fraction whose odd and even parts are limit 1-periodic for |q| > 1, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.
A Convergence Theorem For Continued Fractions Of The Form K_{N=1}^{\Infty}A_{N}/1, James Mclaughlin, Nancy Wyshinski
A Convergence Theorem For Continued Fractions Of The Form K_{N=1}^{\Infty}A_{N}/1, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
In this paper we present a convergence theorem for continued fractions of the form K∞n=1an/1. By deriving conditions on the an which ensure that the odd and even parts of K∞n=1an/1 converge, these same conditions also ensure that they converge to the same limit. Examples will be given.
Ramanujan And The Regular Continued Fraction Expansion Of Real Numbers, James Mclaughlin, Nancy Wyshinski
Ramanujan And The Regular Continued Fraction Expansion Of Real Numbers, James Mclaughlin, Nancy Wyshinski
Mathematics Faculty Publications
In some recent papers, the authors considered regular continued fractions of the form [a0; a, · · · , a | {z } m , a2 , · · · , a2 | {z } m , a3 , · · · , a3 | {z } m , · · · ], where a0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived …
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.