Congruences for Five Triangles

We use the level 12 theory of theta functions to derive hypergeometric transformation formulas and show that forty-two functions are equal. We discuss applications to Ramanujan’s series for 1/π and show how to obtain iterations that converge rapidly to 1/π. CUBIC THETA FUNCTIONS A GLIMPSE S.Bhargava [email protected] Department of Studies in Mathematics University of Mysore Manasagangotri Mysuru 570 006, INDIA. On the six pages pp 257-262 of his second notebook, Ramanujan records statements of several identities, which in fact belong to three theories alternative to classical theories of theta and elliptic functions the most interesting of these theories being the so called cubic theory. Elsewhere, in one of his seminal papers published in 1914, he hints at the existence of such, what he calls there corresponding theories. Further, in one of his letters to Hardy written from a nursing home in England in 1918, he communicates an analytic formulation of a theorem of Dirichlet concerning the number of representations of an integer by a quadratic form. This analytic formulation indeed happens to be the Lambert series representation for the cubic analogue of one of Jacobis special theta functions. In this talk, we will take a brief look at some of the interesting developments that have since taken place concerning just one strand of Ramanujans alternative theories. ICJMSNTSFA, 8th − 10th, AUGUST, 2016, Pondicherry, India. 3 NEW CONGRUENCES FOR PARTITIONS INTO ODD (DISTINCT) PARTS Nayandeep Deka Baruah [email protected] Department Mathematics, Tezpur University, Tezpur, INDIA. Let p0(n) denote the number of partitions of n into odd parts (or, by Euler’s Theorem, into distinct parts). We will present some new(?) congruences for p0(n) modulo 2, 4, and 5. This is a joint work with my students Zakir Ahmed and Nilufar Mana Begum. A STUDY ON APPELL-TYPE CHANGHEE POLYNOMIALS AND THEIR APPLICATIONS Lee-Chae Jang [email protected]. Graduate School of Education, Konkuk University, Seoul 143-701, Republic of Korea. Recently, Lim-Qi have derived integral identities for Appell-type λ-Changhee numbers from the fermionic integral equation (see [24]). The degenerate Bernoulli polynomials, the degenerate version of well-known families of polynomials, were introduced by Carlitz and after that many researchers have studied the degenerate special polynomials(see[1-7,2426]). The goal of this paper is to consider the Appell-type Changhee polynomials, another version of the Changhee polynomials in (3) and derive some properties of these polynomials. Furthermore, we investigate certain identities for those polynomials. 4 ICJMSNTSFA, 8th − 10th, AUGUST, 2016, Pondicherry, India. A NOTE ON SOME IDENTITIES OF THE DEGENERATE FROBENIUS-GENOCCHI POLYNOMIALS Jeong Gon Lee [email protected] Division of Mathematics and Informational Statistics and Nanoscale Science and Technology Institute, Wonkwang University, Iksan 570-749, Republic of Korea. 2010 MSC: 05A10, 11B68, 11S80, 05A19. In recent years, many researchers have studied various types of special polynomials, for examples, Barnes-type degenerate Euler polynomials, the degenerate Frobenius-Euler polynomials, Daehee polynomials, Changhee polynomials, and Boole polynomials etc. Remark that Kim(2015) constructed the degenerate Frobenius-Euler polynomials and numbers and studied some identities of these polynomials. Thus, our motivation in this paper is to define the degenerate Frobenius-Genocchi polynomials and to investigate some new and interesting properties of these polynomials. POLYNOMIAL IDENTITIES ARISING IN TWO PHOTON LASER SCANNING MICROSCOPY Satish Iyengar [email protected] a Statistics Department University of Pittsburgh Pittsburgh, PA USA Photon counting methods can give better images than analog methods in two-photon laser scanning microscopy. However, photon detectors have a dead period that leads to undercounts. Thus, it is useful to derive the distribution of the observed photon counts in order to make inferences about the total emitted photons. In the course of deriving the distribution of the number of observed photons, we encounter several classes of polynomial identities. In particular, we use two different approaches to the problem of computing the distribution to prove one class of identities. This is joint work with Burcin Simsek. ICJMSNTSFA, 8th − 10th, AUGUST, 2016, Pondicherry, India. 5 Q-BERNSTEIN BASES AND THEIR APPLICATIONS Yilmaz Simsek [email protected] Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya, Turkey. 2010 MSC: 65Qxx, 33Dxx, 65D17, 26Cxx, 30C10.