Search

This note discusses elliptic functions in Ramanujan's work.

We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as $$ \int_0^{e^{-2\pi/\sqrt{10}}} q\prod_{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \text{d} q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right). $$ We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.

The diffusionless martensitic phase transition from a hexagonal close-packed (hcp) arrangement to the face-centered close-packed (fcc) and subsequently the body-centered cubic (bcc) lattice is discussed for a Lennard-Jones solid. The associated lattice vectors to construct the underlying bi-lattice for a Burgers-Bain-type of transformation require a minimum of four parameters $(a,\alpha,\beta,\gamma=c/a)$ describing, beside the change in the base lattice parameters $a$ and $c$, the shear force acting on the hexagonal base plane through the parameter $\alpha$, and the sliding force of the middle layer in the original AB hexagonal packing arrangement through the parameter $\beta$. By optimizing the lattice parameters $a$ and $\beta$ for a $(n,m)$-Lennard-Jones potential, we obtain a simple two-dimensional picture for the complete Burgers-Bain-type hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition. From the generalized lattice vectors we were able to construct the corresponding lattice sums in terms of inverse power potentials applying fast converging Bessel function expansions using a Terras decomposition of the Epstein zeta function combined with a Van der Hoff-Benson expansion for the lattice sums. This allows the cohesive energy to be determined to computer precision for a Lennard-Jones solid. For six different combinations of $(n,m)$-Lennard-Jones potentials the energy $(\alpha,\gamma)$ hypersurface was then mapped out and studied in more detail. We show that for a Lennard-Jones model the minimum energy path is found to be a two-step hcp$\rightarrow$fcc$\rightarrow$bcc transition process. The lowest transition state in each case can be regarded as an upper limit to a hypothetical true minimum energy path out of the many possibilities in a hcp$\leftrightarrow$fcc$\leftrightarrow$bcc phase transition.

The Ap\'ery numbers may be defined by a cubic three-term recurrence relation, that is, a three-term relation where the coefficients are polynomials in the index of degree $3$. In this work, we first provide a systematic review of Ap\'ery numbers and other related sequences that satisfy quadratic or cubic three-term recurrence relations, and show how they are interrelated and how they may be classified. This leads to sequences defined by cubic $k$-term recurrence relations. The cases corresponding to $k=2$ in this framework lead to Ramanujan's theories of elliptic functions to alternative bases, while the cases corresponding to $k=3$ correspond to the Ap\'ery, Domb, Almkvist--Zudilin numbers and other sequences that are well-studied. We conduct a detailed analysis for the case $k=4$. Some of the sequences that arise are new. Of particular interest are ten sequences that are said to be self-starting in the sense that a single initial condition is enough to start the recurrence relation. Of additional interest are two sequences which take values in $\mathbb{Z}[i]$ and two others with values in $\mathbb{Z}[\sqrt{2}]$. Congruence properties and asymptotic expansions for the ten self-starting sequences are investigated and several conjectures are presented. For example, we conjecture that the integer-valued sequence defined by the recurrence relation \begin{align*} (n+1)^3T(n+1) &=2(2n+1)(5n^2+5n+2)T(n) \\ &\qquad -8n(7n^2+1)T(n-1)+22n(2n-1)(n-1)T(n-2) \end{align*} and initial condition $T(0)=1$ satisfies a Lucas congruence for every prime $p$. Moreover, the sequence is conjectured to satisfy the supercongruence $$ T(pn) \equiv T(n) \pmod{p^2} \quad\text{for all positive integers $n$} $$ if $p=2,\;59$ or $5581$, and for no other primes $p<10^4$.

We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and representations of sums $r_N(m)$ of squares for $N$-dimensional Madelung constants, $M_N(s)$, where $s$ is the exponent of the Madelung series (usually chosen as $s=1/2$). The functional behavior including analytical continuation, and the convergence of the Bessel function expansion is discussed in detail. Recursive definitions are used to evaluate $r_N(m)$. Values for $M_N(s)$ for $s=\tfrac{1}{2}, \tfrac{3}{2}, 3$ and 6 for dimension up to $N=20$ and for $M_N(1/2)$ up to $N=100$ are presented. Zucker's original analysis on $N$-dimensional Madelung constants for even dimensions up to $N=8$ and their possible continuation into higher dimensions is briefly analyzed., Comment: 16 pages, 7 figures

A smooth path of rearrangement from the body-centered cubic (bcc) to the face-centered cubic (fcc) lattice is obtained by introducing a single parameter to cuboidal lattice vectors. As a result, we obtain analytical expressions in terms of lattice sums for the cohesive energy. This is described by a Lennard-Jones (LJ) interaction potential and the sticky hard sphere (SHS) model with an $r^{-n}$ long-range attractive term. These lattice sums are evaluated to computer precision by expansions in terms of a fast converging series of Bessel functions. Applying the whole range of lattice parameters for the SHS and LJ potentials demonstrates that the bcc phase is unstable (or at best metastable) toward distortion into the fcc phase. Even if more accurate potentials are used, such as the extended LJ potential for argon or chromium, the bcc phase remains unstable. This strongly indicates that the appearance of a low temperature bcc phase for several elements in the periodic table is due to higher than two-body forces in atomic interactions., Comment: 10 pages, 5 figures

Lattice sums of cuboidal lattices, which connect the face-centered with the mean-centered and the body-centered cubic lattices through parameter dependent lattice vectors, are evaluated by decomposing them into two separate lattice sums related to a scaled cubic lattice and a scaled Madelung constant. Using theta functions we were able to derive fast converging series in terms of Bessel functions. Analytical continuations of these lattice sums are discussed in detail.

We record $$ \binom{42}2+\binom{23}2+\binom{13}2=1192 $$ functional identities that, apart from being amazingly amusing by themselves, find applications in derivation of Ramanujan-type formulas for $1/\pi$ and in computation of mathematical constants., Comment: 31 pages

The Ramanujan--Mordell Theorem for sums of an even number of squares is extended to other quadratic forms and quadratic polynomials.

Special arithmetic series $f(x)=\sum_{n=0}^{\infty}c_nx^n$, whose coefficients $c_n$ are normally given as certain binomial sums, satisfy "self-replicating" functional identities. For example, the equation $$\frac1{(1+4z)^2}f\biggl(\frac{z}{(1+4z)^3}\biggr)=\frac1{(1+2z)^2}f\biggl(\frac{z^2}{(1+2z)^3}\biggr)$$ generates a modular form $f(x)$ of weight 2 and level 7, when a related modular parametrization $x=x(\tau)$ is properly chosen. In this note we investigate the potential of describing modular forms by such self-replicating equations as well as applications of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms for computing $\pi$ and other related constants. Finally, we indicate some possibilities to extend the functional equations to a two-variable setting., Comment: 16 pages

We adopt the "translation" as well as other techniques to express several identities conjectured by Z.-W. Sun in arXiv:1102.5649v47 by means of known formulas for $1/\pi$ involving Domb and other Ap\'ery-like sequences., Comment: 23 pages