The products of three theta functions and the general cubic theta functions

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan＇s identities, Winquist＇s identity and many other interesting identities.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower hounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy‘s Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Partitions with initial repetitions

A variety of interesting connections with modular forms, mock theta functions and Rogers- Ramanujan type identities arise in consideration of partitions in which the smaller integers are repeated as summands more often than the larger summands. In particular, this concept leads to new interpretations of the Rogers Selberg identities and Bailey＇s modulus 9 identities.     

EXACT EVALUATIONS OF FINITE TRIGONOMETRIC SUMS BY SAMPLING THEOREMS

We use the sampling representations associated with Sturm-Liouville difference operators to derive generalized integral-valued trigonometric sums. This extends the known results where zeros of Chebyshev polynomials of the first kind are involved to the use of the eigenvalues of difference operators, which leads to new identities. In these identities Bernoulli’s numbers play a role similar to that of Euler’s in the old ones. Our technique differs from that of Byrne-Smith (1997) and Berndt-Yeap (2002).     

On representations of real Jacobi groups

We consider a category of continuous Hilbert space representations and a category of smooth Fr’echet representations,of a real Jacobi group G.By Mackey’s theory,they are respectively equivalent to certain categories of representations of a real reductive group L.Within these categories,we show that the two functors that take smooth vectors for G and for L are consistent with each other.By using Casselman-Wallach’s theory of smooth representations of real reductive groups,we define matrix coefficients for distributional vectors of certain representations of G.We also formulate Gelfand-Kazhdan criteria for real Jacobi groups which could be used to prove multiplicity one theorems for Fourier-Jacobi models.     

递推数列的非线性表达式的构造

A type of nonlinear expressions of Lucas sequences are established inspired by Hsu [A nonlinear expression for Fibonacci numbers and its consequences.J.Math.Res.Appl.,2012,32(6):654–658].Using the relationships between the Lucas sequence and other linear recurring sequences satisfying the same recurrence relation of order 2,i.e.,the Horadam sequences,we may transfer the identities of Lucas sequences to the latter.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy’s Theorem for Jacobi transform by using the sharp estimate for the heat kernel.     

SOME NEW IDENTITIES OF ROGERS-RAMANUJAN TYPE

In this paper,we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz’s inversions formulas and Jackson s transformation formula.In terms of application,by specializing certain parameters in the two transformations,four Rogers-Ramanujan type identities associated with moduli 20 are obtained.     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

Partition identities arising from theta function identities

The authors show that certain theta function identities of Schroeter and Ramanujan imply elegant partition identities.     

The multisection method for triple products and identities of Rogers-Ramanujan type

By applying the bisection and trisection method to Jacobi's triple product identity, we establish several identities factorizing sum and difference of infinite products, which lead, in turn, to new and elementary proofs for twenty identities of Rogers-Ramanujan type.     

Jacobi's Two-Square Theorem and Related Identities

We show that Jacobi's two-square theorem is an almost immediate consequence of a famous identity of his, and draw combinatorial conclusions from two identities of Ramanujan.     

A note on a Jacobian identity

An identity involving eight-fold infinite products, first derived by Jacobi in his theory of theta functions, is the subject of this note. Three similar identities, including one that implies Jacobi's identity, are presented.

     

Circular summation of theta functions in Ramanujan's Lost Notebook

In this paper, we prove Ramanujan's circular summation formulas previously studied by S.S. Rangachari, S.H. Son, K. Ono, S. Ahlgren and K.S. Chua using properties of elliptic and theta functions. We also derive identities similar to Ramanujan's summation formula and connect these identities to Jacobi's and Dixon's elliptic functions. At the end of the paper, we discuss the connection of our results with the recent thesis of E. Conrad.     

New polynomial analogues of Jacobi''s triple product and Lebesgue''s identities

In a recent paper by the authors, a bounded version of Göllnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of Jacobi and Lebesgue. We also derive a two parameter extension of Jacobi's famous triple product identity.     

Infinite Families of Exact Sums of Squares Formulas,Jacobi Elliptic Functions,Continued Fractions,and Schur Functions

In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to 4n ² or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan's tau function (n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the -function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n ² + n, 2n ² – n that appear in Macdonald's work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing a positive integer by sums of 4n ² or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's C nonterminating ₆₅ summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's 2, 4, 6, and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n ² and n(n + 1) squares. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Sierpinski (1907), Uspensky (1913, 1925, 1928), Bulygin (1914, 1915), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Bell (1919), Estermann (1936), Rankin (1945, 1962), Lomadze (1948), Walton (1949), Walfisz (1952), Ananda-Rau (1954), van der Pol (1954), Krätzel (1961, 1962), Bhaskaran (1969), Gundlach (1978), Kac and Wakimoto (1994), and, Liu (2001). We list these authors by the years their work appeared.     

Approximation properties of Jacobi's algorithm

The paper improves a previous result which is a generalization of Hurwitz' theorem for Jacobi's algorithm.     

The use of jacobi's lemma in unimodularity theory

The paper applies Jacobi's fundamental result on minors of the adjoint matrix to obtain properties on determinants of unimodular matrices.     

Approximation properties of Jacobi's algorithm

The paper improves a previous result which is a generalization of Hurwitz' theorem for Jacobi's algorithm.     

Durfee rectangles and the Jacobi triple product identity

By means of the Durfee rectangles of partitions, a very elementary proof for Jacobi's triple product identity and its finite analogue is presented.