Products Of Binomial Coefficients Research Articles

A remark on an identity involving products of binomial coefficients

A remark on an identity involving products of binomial coefficients

Computation of several Hessenberg determinants

Abstract In the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.

Evaluations of sums involving harmonic numbers and binomial coefficients

ABSTRACTIn this paper, by the Faà di Bruno formula, we establish the decompositions of two general fractions involving the reciprocals of products of binomial coefficients. Using the decompositions, we discuss the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as and some other forms. We present some explicit evaluations as examples and provide the Maple package to compute the sums and . It can be found that this work gives a unified approach to such sums and generalizes many known results in the literature.

Products of binomial coefficients and unreduced Farey fractions

This paper studies the product [Formula: see text] of the binomial coefficients in the [Formula: see text]th row of Pascal’s triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order [Formula: see text]. It studies its size as a real number, measured by [Formula: see text], and its prime factorization, measured by the order of divisibility [Formula: see text] by a fixed prime [Formula: see text], each viewed as a function of [Formula: see text]. It derives three formulas for [Formula: see text], two of which relate it to base [Formula: see text] radix expansions of integers up to [Formula: see text], and which display different facets of its behavior. These formulas are used to determine the maximal growth rate of each [Formula: see text] and to explain structure of the fluctuations of these functions. It also defines analogous functions [Formula: see text] for all integer bases [Formula: see text] using base [Formula: see text] radix expansions replacing base [Formula: see text]-expansions. A final topic relates factorizations of [Formula: see text] to Chebyshev-type prime-counting estimates and the prime number theorem.

The Explicit Identities for Spectral Norms of Circulant-Type Matrices Involving Binomial Coefficients and Harmonic Numbers

The explicit formulae of spectral norms for circulant-type matrices are investigated; the matrices are circulant matrix, skew-circulant matrix, andg-circulant matrix, respectively. The entries are products of binomial coefficients with harmonic numbers. Explicit identities for these spectral norms are obtained. Employing these approaches, some numerical tests are listed to verify the results.

Fibonacci polynomials: compositions and cyclic products

Abstract Fibonacci (alias Chebyshev) polynomials enjoy particular composition properties. These can be seen (and proved) from a combinatorial perspective by interpreting these polynomials as matching polynomials. An enumerative technique for cyclic structures is applied to obtain a generating polynomial identity for cyclic products of binomial coefficients in terms of matching polynomials.

Some composition determinants

We compute several parametric determinants in which rows and columns are indexed by compositions, where the entries are either products of binomial coefficients or products of powers. These results generalize previous determinant evaluations due to the first and fourth author [SIAM J. Matrix Anal. Appl. 23 (2001) 459–471] and [A polynomial generalization of the power-compositions determinant, Linear Multilinear Algebra, in press], and they prove two conjectures of the second author [Advanced determinant calculus: a complement, preliminary version].

Difference of Sums Containing Products of Binomial Coefficients and Their Logarithms

Properties of the difference of two sums containing products of binomial coefficients and their logarithms which arise in the application of Shannon's information theory to a certain class of covert channels are deduced. Some allied consequences of the latter are also recorded.

Universal factorization of 3n-j(j > 2) symbols of the first and second kinds for SU(2) group and their direct and exact calculation and tabulation

We show that general $3n-j (n>2)$ symbols of the first kind and the second kind for the group SU(2) can be reformulated in terms of binomial coefficients. The proof is based on the graphical technique established by Yutsis, et al. and through a definition of a reduced $6-j$ symbol. The resulting $3n-j$ symbols thereby take a combinatorial form which is simply the product of two factors. The one is an integer or polynomial which is the single sum over the products of reduced $6-j$ symbols. They are in the form of summing over the products of binomial coefficients. The other is a multiplication of all the triangle relations appearing in the symbols, which can also be rewritten using binomial coefficients. The new formulation indicates that the intrinsic structure for the general recoupling coefficients is much nicer and simpler, which might serves as a bridge for the study with other fields. Along with our newly developed algorithms, this also provides a basis for a direct, exact and efficient calculation or tabulation of all the $3n-j$ symbols of the SU(2) group for all range of quantum angular momentum arguments. As an illustration, we present teh results for the $12-j$ symbols of the first kind.

An Amusing Representation of x sinx

This proof was motivated by the classical one of the Ramanujan congruence mod7, i.e., p(7m + S) -O (mod7). In that case one multiplies P_3(x) by P7(x). Modulo 7, one gets P4(x) -t + xu + X3U, where t, u, u are all power series in X7. Squaring this gives P8(x) having coefficients of x7m+5 divisible by 7, which again won't change upon multiplying by P_7(x). Alternate proofs can be obtained by replacing the cubing in the modS case and the squaring in the mod 7 case by taking the square roots and fourth roots, respectively, of the formal power series. This gives the function P(x) directly; however, there must now be an additional argument to show that the binomial coefficients generated (or the products of binomial coefficients in the mod 7 case) are O in the appropriate modulus.

The Jordan Normal Form of Matrices with Products of Binomial Coefficients as Entries

The Jordan Normal Form of Matrices with Products of Binomial Coefficients as Entries

A Finite Sum of Products of Binomial Coefficients (C. C. Grosjean)

A Finite Sum of Products of Binomial Coefficients (C. C. Grosjean)

New algorithms for calculating 3n-j symbols

Compact expressions are presented for the 3n-j symbols, where 1 ⩽ n ⩽ 4, which feature sums over products of binomial coefficients, and certain integer triangular coefficients. The triangular coefficients in turn can be expressed as products of binomial coefficients. Thus in the formulas presented for the 3n-j symbols, the dependence on numerous factorials, formally as well as computationally, has been completely eliminated. While formulas which incorporate summations over products of binomial coefficients have been known for the 3- j and 6- j symbols, the introduction of the triangular coefficients, and the application of the binomial/triangular scheme to 3n-j symbols with n > 2, provide important new results. The new formulas are simpler, and they permit more efficient computations of the 3n-j symbols, both in exact and in floating point format, than most schemes which are currently in use. © 1993 John Wiley & Sons, Inc.

A Finite Sum of Products of Binomial Coefficients

A Finite Sum of Products of Binomial Coefficients

On Congruences Involving Sums of Products of Binomial Coefficients

On Congruences Involving Sums of Products of Binomial Coefficients