Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type,
2010
Northern Illinois University
Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi
Mathematics Faculty Publications
We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive m-versions of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such m-versions of Rogers Ramanujan-type identities.
An Identity Motivated By An Amazing Identity Of Ramanujan,
2010
West Chester University of Pennsylvania
An Identity Motivated By An Amazing Identity Of Ramanujan, James Mclaughlin
Mathematics Faculty Publications
Ramanujan stated an identity to the effect that if three sequences {an}, {bn} and {cn} are defined by r1(x) =: ∑∞ n=0 anx n , r2(x) =: ∑∞ n=0 bnx n and r3(x) =: ∑∞ n=0 cnx n (here each ri(x) is a certain rational function in x), then a 3 n + b 3 n − c 3 n = (−1)n , ∀ n ≥ 0. Motivated by this amazing identity, we state and prove a more general identity involving eleven sequences, the new identity being ”more general” in the sense that equality holds not just for the power …
Some New Transformations For Bailey Pairs And Wp-Bailey Pairs,
2010
West Chester University of Pennsylvania
Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin
Mathematics Faculty Publications
We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.
Some Applications Of A Bailey-Type Transformation,
2010
West Chester University of Pennsylvania
Some Applications Of A Bailey-Type Transformation, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
If k is set equal to aq in the definition of a WP Bailey pair, βn(a, k) = Xn j=0 (k/a)n−j (k)n+j (q)n−j (aq)n+j αj (a, k), this equation reduces to βn = Pn j=0 αj . This seemingly trivial relation connecting the αn’s with the βn’s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for poly-basic hypergeometric series.
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem,
2010
Claremont McKenna College
Ergodic And Combinatorial Proofs Of Van Der Waerden's Theorem, Matthew Samuel Rothlisberger
CMC Senior Theses
Followed two different proofs of van der Waerden's theorem. Found that the two proofs yield important information about arithmetic progressions and the theorem. van der Waerden's theorem explains the occurrence of arithmetic progressions which can be used to explain such things as the Bible Code.
The Fibonacci Sequence,
2010
Parkland College
The Fibonacci Sequence, Arik Avagyan
A with Honors Projects
A review was made of the Fibonacci sequence, its characteristics and applications.
