Triple Sum Research Articles

Triple Symmetric Sums of Circular Binomial Products

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics.

Rogers–Ramanujan type identities involving double, triple and quadruple sums

Rogers–Ramanujan type identities involving double, triple and quadruple sums

An efficient polar cubic equation of state for predictive modeling of phase behavior and critical phenomena of mixtures

AbstractA polar cubic equation of state (EOS) is developed by incorporating the dipolar theory of Jog and Chapman into the Soave‐Redlich‐Kwong (SRK) EOS. We propose simplifying assumptions in the dipolar term of Jog and Chapman to reduce the double and triple sums in the theory to single sums. The simplified version of the dipolar theory can significantly improve computational speed and can be used with either Cubic EOS or SAFT‐based EOS. The proposed model, which we call Simplified Polar SRK (SP‐SRK), is parametrized in a similar fashion to classical cubic EOS to exactly reproduce , and will self‐consistently reduce to the base SRK EOS in the absence of polar interactions. Binary VLE data with a non‐polar reference hydrocarbon is used to extract the polarity of the respective functional group. The model shows superior performance in capturing the phase behavior of polar mixtures compared to the base SRK.

Double and triple character sums and gaps between the elements of subgroups of finite fields

For an odd prime [Formula: see text], let [Formula: see text] be the finite field of [Formula: see text] elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any [Formula: see text], we also obtain new upper bounds of the following double character sum [Formula: see text] and a triple character sum [Formula: see text] with [Formula: see text] and multiplicative subgroups [Formula: see text] of order [Formula: see text] and [Formula: see text], respectively.

Sparse sets that satisfy the prime number theorem

For arbitrary real t>1 we examine the set {⌊x/nt⌋:n≤x}. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set.

Triple correlation sums of coefficients of cuspidal forms

Triple correlation sums problem concerns the non-trivial power-saving bounds for the correlation of three objects. It is conjectured that these sums are non-trivial in any fixed but arbitrarily given ranges. In this paper, the uniform non-trivial bounds for the triple correlation sums ∑m≥1,n≥1λπ(1,m)λ⋆(n)λf(m+pn)U(m/X)V(n/H) in the level aspect are derived, where π is any GL3-Maaß cuspidal form, f∈Bk⁎(p), any Hecke newform of prime level p and weight k∈N+, and λ⋆(n), n≥1, are certain coefficients of arithmetic interest. As a result, we show that sums of this type follow the 1/3-significance level. We study the strength of the result by specifying ⋆ being the cusp forms on GL3 and GL2, respectively, and further obtain the more significant cancellations in these sums.

Multi-sum Rogers-Ramanujan type identities

We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are still infinite products. The method used here is motivated by Rosengren's proof of the Kanade-Russell identities.

Q-Supercongruences on Triple and Quadruple Sums

Inspired by the recent work of El Bachraoui, we present some new q-supercongruences on triple and quadruple sums of basic hypergeometric series. In particular, we give a q-supercongruence modulo the fifth power of a cyclotomic polynomial, which is a q-analogue of the quadruple sum of Van Hamme’s supercongruence (G.2).

On strongly walk regular graphs, triple sum sets and their codes

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues k> theta _1> theta _2 > theta _3, and that the triple (theta _1, theta _2, theta _3) satisfies a certain homogeneous polynomial equation of degree s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for s = 5 and s = 7, there are only the obvious solutions, and we conjecture this to remain true for all (odd) s ge 9.

Generating Functions for Four Classes of Triple Binomial Sums

By means of the generating function approach, four classes of triple sums involving circular products of binomial coefficients are investigated. Recurrence relations and rational generating functions are established.

A New Result in Form of Finite Triple Sums for a Series from Ramanujan’s Notebooks

We consider a function g(r,x,u) with x,u∈ℂ and r∈ℕ, which, over a symmetric domain, equals the sum of an infinite series as noted in the 16th Entry of Chapter 3 in Ramanujan’s second notebook. The function attracted new attention since it was established to be closely connected to the theory of labelled trees. However, to the best of our knowledge, a closed-form solution allowing, e.g., the rapid computation of g(r,x,u) in Mathematica without explicit use of recursions has been lacking until now. Our proposed formula transforms the part depending on the variable u into a more symmetric form, which then appears inside a finite triple sum consisting of binomials and Stirling numbers of the second kind.

Parametrization of Kloosterman sets and SL3-Kloosterman sums

We stratify the SL3 big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the SL3 long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup Γ0(N)⊆SL3(Z). Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.

Hecke-type triple sums associated with mock theta functions

Hecke-type triple sums associated with mock theta functions

Robust Asynchronous H∞ Observer-Based Control Design for Discrete-Time Switched Singular Systems with Time-Varying Delay and Sensor Saturation: An Average Dwell Time Approach

This work discuss the robust stabilization problem for discrete-time switched singular systems with simultaneous presence of time-varying delay and sensor nonlinearity. To this end, an observer-based controller was synthesized that works under asynchronous switching signals. Investigating the average dwell time approach and using a Lyapunov–Krasovskii functional with triple sum terms, sufficient conditions were derived for achieving the existence of such asynchronous controller and guaranteeing the resulting closed-loop system to be exponentially admissible with H∞ performance level. Subsequently, the effectiveness of the proposed control scheme was verified through two numerical examples.

Triple Equal Sum Labeling of Some Graphs

Triple Equal Sum Labeling of Some Graphs

Residue Sum Formula for Pricing Options under the Variance Gamma Model

We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties of multidimensional complex analysis, with particular emphasis on the multidimensional Jordan Lemma and the application of residue calculus to a Mellin–Barnes integral representation in C3, for the call option price. Moreover, we derive triple sum series formulas for some of the Greeks associated to the call option and we discuss the numerical accuracy and convergence of the main pricing formula.

Creative Telescoping on Multiple Sums

We showcase a collection of practical strategies to deal with a problem arising from an analysis of integral estimators derived via quasi-Monte Carlo methods. The problem reduces to a triple binomial sum, thereby enabling us to open up the holonomic toolkit, which contains tools such as creative telescoping that can be used to deduce a recurrence satisfied by the sum. While applying these techniques, a host of issues arose that partly needed to be resolved by hand. In other words, no creative telescoping implementation currently exists that can resolve all these issues automatically. Thus, we felt the need to compile the different strategies we tried and the difficulties that we encountered along the way. In particular, we highlight the necessity of the certificate in these computations and how its complexity can greatly influence the computation time.

Triple sums including Fibonacci numbers with three binomial coefficients

In this paper, we consider some triple sums that involve Fibonacci numbers with three binomial coefficients. We chose the indices of Fibonacci numbers as linear combination of the summation indices. Moreover, various types of alternating analogues of them whose powers depend on the index or indices are computed. These sums are evaluated in nice multiplication forms in terms of Fibonacci and Lucas numbers.

Triple electron emission from surfaces: Energy and angle relations

We discuss a proof-of-principle experiment in which we detect triple electron emission from a surface due to primary electron impact. The new aspect is the ability to record the energies and emission directions of the ejected electrons. We selected NiO films as a target, which have shown in previous electron pair emission studies to give an enhanced intensity compared to other materials. The triple sum energy spectrum displays a shape consistent with a self-convolution of the electronic density of states. We define two different emission geometries. While the energy distributions are essentially identical, the intensity levels differ by a factor of 2. Imposing a geometrical constraint on one of the emitted triples shows that the available energy is equally shared among the other two electrons. We discuss our findings within a simplified scattering model. We also present angular distributions. Prominent intensity minima for electron emission in the same direction are not observed in contrast to our previous electron pair emission studies.

Seven-point conformal blocks in the extended snowflake channel and beyond

Seven-point functions have two inequivalent topologies or channels. The comb channel has been computed previously and here we compute scalar conformal blocks in the extended snowflake channel in $d$ dimensions. Our computation relies on the known action of the differential operator that sets up the operator product expansion in embedding space. The scalar conformal blocks in the extended snowflake channel are obtained as a power series expansion in the conformal cross-ratios whose coefficients are a triple sum of the hypergeometric type. This triple sum factorizes into a single sum and a double sum. The single sum can be seen as originating from the comb channel and is given in terms of a ${}_3F_2$-hypergeometric function, while the double sum originates from the snowflake channel which corresponds to a Kamp\'e de F\'eriet function. We verify that our results satisfy the symmetry properties of the extended snowflake topology. Moreover, we check that the behavior of the extended snowflake conformal blocks under several limits is consistent with known results. Finally, we conjecture rules leading to a partial construction of scalar $M$-point conformal blocks in arbitrary topologies.