Papers by Victor H. Moll
If you would like a hard copy, please email me at vhm at math.tulane.edu
- The $p$-adic valuation of Stirling
numbers (with A. Berrizbeitia, L. Medina, A. Moll and L. Noble)
(20 pages).
We present results on the p-adic valuation of
S(n,k).
Two types of trees describing this valuation are introduced.
A conjecture descring a branching phenomena for these trees is presented.
The Mathematica package containing the algorithm and examples is in
here.
- Wallis-Ramanujan-Schur-Feynman
(with T. Amdeberhan, O. Espinosa and A. Straub)((13 pages).
To appear in American Mathematical Monthly
The evaluation of an integral of a rational function is given in terms of
Schur functions. This example
generalizes Wallis classical example and some particular cases were
described by Ramanujan.
A connection with the evaluation of certain sums appearing in Feynman diagrams is presented.
- The integrals in Gradshteyn and Ryzhik. Part 14: An elementary evaluation of entry 3.411.5
(with Tewodros Amdeberhan) (7 pages).
We provide an elementary evaluation of a series that gives entry 3.411.5 in the table of integrals
by I. M. Gradshteyn and Ryzhik. It involves a special value of the dilogarithm function.
- The integrals in Gradshteyn and Ryzhik. Part 13:
Trigonometric forms of the beta function
(6 pages).
We present the evaluation of some integrals in the table of Gradhsteyn and Ryzhik
where the integrand involves trigonometric functions and the result is
expressed in terms
of the beta function.
- Closed-form evaluation of integrals appearing in positronium decay
(with T. Amdeberhan and A. Straub)
(5 pages).
Definite integrals in positronium decay are evaluated using dilogarithms.
To appear in Journal of Math. Physics
- The Cauchy-Schlomilch transformation
(with T. Amdeberhan, L. Glasser, M. C. Jones, R. Posey and D. Varela)
(20 pages).
Definite integrals are evaluated using a simple transformation derived by
Cauchy
and popularized by Schlomilch.
- On some families of integrals connected to the Hurwitz zeta function
(with Alexander Patkowski) (7 pages).
We provide the evaluation of several definite integrals involving the
kernel 1/(exp(t)-1) and its variations in terms of the Hurwitz zeta function.
- The integrals in Gradshteyn and Ryzhik. Part 12:
Some logarithmic integrals
(with R. Posey) (8 pages).
Scientia, Series A: Math. Sciences 18, 2009, 77-84.
We present the evaluation of some integrals in the table of Gradhsteyn and Ryzhik
where the integrand is the product of a rational function with comples poles and the logarithm function.
- A fast numerical algorithm for the integration of rational functions. (with Dante Manna, Luis Medina and Armin Straub)
(18 pages)
The rational Landen transformations are used to develop a fast numerical
algorithm for the integration of a rational function on the
whole line.
The Mathematica package containing the algorithm and examples is in
here. Examples are contained in
here.
- Seized opportunities. (14 pages)
We describe some of the Mathematical issues involved in the
evaluation of definite integrals.
To appear in Notices Amer. Math. Soc.
- The p-adic valuations of sequences counting alternating sign matrices. (with Xinfu Sun) (24 pages)
We describe the p-adic valuation of a formula giving the number of size n
matrices formed
by 0, +1 and -1 with column and row sum equal to 1 and alternating non-zero entries.
Journal of Integer Sequences Vol. 12 (2009), Article 09.3.8
- Definite integrals by the method of brackets. Part 1. (with Ivan Gonzalez) (33 pages)
To appear in Advances in Applied Mathematics
We introduce a new method for the evaluation of definite integrals.
The method is heuristic and it appeared in the context of
integrals coming from Feynman diagrams.
- A remarkable sequence of integers (with Dante Manna) (20 pages)
The evaluation of a quartic integral produce a sequence of rational numbers
with many interesting properties.
The denominators are simple to describe,
their numerators are the integers in the title. Many open problems are
presented.
Expositiones Mathematicae
27, 2009, 289-312.
- The evaluation of Tornheim double sums, Part 2 (with Olivier Espinosa) (38 pages)
To appear in The Ramanujan Journal
Explicit formula for Tornheim double series are given in terms of integrals
involving
the Hurwitz zeta function.
The Mathematica package to evaluate the Tornheim sums is available
here.
- The $p$-adic valuation of $k$-central binomial coefficients
(with Armin Straub and Tewodros Amdeberhan) (11 pages).
We study a sequence of coefficients in the expansion of $(1 - k^2x)^{-1/k}$.
The motivation comes from a problem proposed by H. Montgomery and H. S. Shapiro
in the American Mathematical Monthly, August-September 2008.
To appear in Acta Arithmetica
- The action of Hecke operators on hypergeometric functions
(with Sinai Robins and Kirk Soodhalter) (22 pages).
We study the spectrum of Hecke operators acting on hypergeometric functions.
Polylogarithms appear as eigenfunctions.
- A binary tree representation for the 2-adic valuation of a sequence arising from a rational integral
(with Xinyu Sun) (9 pages).
We provide a binary tree that encodes the 2-adic valuation of the
coefficients d_{l,m} coming from a rational integral.
- A class of logarithmic integrals
(with L. Medina) (30 pages).
Ramanujan Journal 20, 2009, 91-126.
We provide explicit closed-form expressions for integrals of the form
Q(x) log log 1/x where Q is a rational
function.
- A dozen integrals: Russel style
(with T. Amdeberhan) (2 pages).
On June 15, 1876, the Proceedings of the Royal Society of London published a one
page paper by W. H. L. Russell that simply contained 12 definite integrals.
We present 12 integrals that hope will be of interest.
Ramanujan Mathematics Newsletter 18(1), 2008, 7-8.
- The integrals in Gradshteyn and Ryzhik. Part 11:
The incomplete beta function
(with K. Boyadzhiev and L. Medina) (15 pages).
Scientia, Series A: Math. Sciences 18, 2009, 61-75.
We present the evaluation of some integrals in the table of Gradhsteyn and Ryzhik
that are involve the incomplete beta function.
- Asymptotic valuations of sequences satisfying first order
recurrences
(with T. Amdeberhan and L. Medina) (5 pages).
Proc. Amer. Math. Soc. 137, 2009, 885-890.
We discuss the asymptotic behavior of the p-adic valuations of sequences defined by first order
recurrences.
- The integrals in Gradshteyn and Ryzhik. Part 10:
The digamma function
Scientia, Series A: Math. Sciences 17, 2009, 45-66.
(with L. Medina) (22 pages).
We present the evaluation of some integrals in the table of Gradhsteyn and Ryzhik
that are involve the digamma function ( the logarithmic derivative of the gamma function).
- The integrals in Gradshteyn and Ryzhik. Part 9:
Combinations of logarithms, rational and trigonometric functions
(with T. Amdeberhan, J. Rosenberg, A. Straub and P. Whitworth) (18 pages).
Scientia, Series A: Math. Sciences 17, 2009, 27-44.
We present the evaluation of some integrals in the table of Gradhsteyn and Ryzhik
that are combinations of logarithms and rational functions. Some trigonometric
versions of the integrals are also presented.
- A formula for a quartic integral: a survey of old
proofs and some new ones
(with T. Amdeberhan) (10 pages).
Ramanujan Journal 18, 2009, 91-102.
We present a survey of proofs of the evaluation of a
quartic integral and give a new proof using
Landen transformations.
- Solution to Problem 11299
(with T. Amdeberhan) (1 page).
We present a solution to Problem 11299 by Pablo Fernandez Refolio.
- An iterative method for numerical
integration of rational functions
(with D. Manna) (14 pages).
Contemporary Mathematics, 471, 2008, 161-174.
We employ the rational Landen transformations to produce a numerical method
for the integration of rational functions.
- The Laplace transform of the digamma function: an integral due to Glasser, Manna and Oloa
(with T. Amdeberhan and O. Espinosa) (10 pages).
Proc. Amer. Math. Soc. 136, 2008, 3211-3221.
We present an analytic expression for the Laplace transform of the
digamma function.
- The integrals in Gradshteyn and Ryzhik. Part 8:
Combinations of powers, exponentials and logarithms
(with A. Straub, J. Rosenberg and P. Whitworth) (10 pages).
Scientia, Series A: Math. Sciences 16, 2008, 41-50.
We present the evaluation of some integrals in the table of Gradhsteyn and Ryzhik
that are combinations of powers, exponentials and logarithms.
- The integrals in Gradshteyn and Ryzhik. Part 7:
Elementary examples
(with T. Amdeberhan) (14 pages).
Scientia, Series A: Math. Sciences 16, 2008, 25-40.
We present some elementary integrals that appear in the table of Gradhsteyn and Ryzhik.
- The integrals in Gradshteyn and Ryzhik. Part 6:
The beta function
(14 pages).
Scientia, Series A: Math. Sciences 16, 2008, 9-24.
We present some elementary integrals that can be expressed in terms of the
beta function.
- The $2$-adic valuation of Stirling
numbers (with T. Amdeberhan
and D. Manna) (23 pages).
Experimental Mathematics, 17, 2008, 69-82.
We present results on the 2-adic valuation of
S(n,k).
A conjecture for an explicit formula for this valuation is proved
in the case k=5.
Interesting pictures for k greater than 5.
- Landen survey
(with D. Manna) (28 pages).
We present a summary of Landen transformations and the
corresponding arithmetic-geometric means.
Probability, Geometry and Integrable Systems
MSRI Publications 55, 2008, 287-319
Special volume honoring Henry McKean.
The whole volume can be seen here
- Integrales definidas: Analisis, Numeros y Experimentos
Revista Cientifica Tumbaga 2, 2008, 138-174.
Notes in Spanish from a course given at the "Semana de la Matematica" in
Valparaiso, Chile.
- Arithmetical properties of a sequence arising from an arctangent sum
Journal of Number Theory
128, 2008, 1808-1847 (with T. Amdeberhan and L. Medina).
The sequence x[n] = (n + x[n-1])/(1- n x[n-1]) is discussed. We prove that
x[n] is not zero for n > 5 and conjecture
that x[n] is not an integer for
n > 4. Conjecture 1.5 (that the product of terms 1 + j^2 is not a square) has been proved
by Javier Cilleruelo. A preprint is available here.
- The $2$-adic valuation of a sequence
arising from a rational integral
(with T. Amdeberhan
and D. Manna) ( 13 pages).
Journal of Combinatorial Theory, Series A, 115, 2008, 1474-1486.
We establish an algorithm to compute the 2-adic valuation of a
sequence of coefficients that have appeared
in the evaluation
of a rational integral.
A combinatorial interpretation is clarified.
- An elementary trigonometric equation
College Mathematical Journal 39, 2008, 394-398.
We present some solutions of the equation A tan t + B sin t = C with
A, B and C^2 rational numbers.
- The integrals in Gradshteyn and Ryzhik. Part 5:
Some trigonometric integrals
(with T. Amdeberhan and L. Medina).
Scientia, Series A: Math. Sciences 15, 2007, 47-60.
We present the evaluations of definite integrals involving x^p times
integer powers of cos x.
- The integrals in Gradshteyn and Ryzhik. Part 4:
The gamma function
Scientia, Series A: Math. Sciences 15, 2007, 37-46.
We present some elementary integrals that can be expressed in terms of the
gamma function.
- The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of
logarithms and exponentials
Scientia, Series A: Math. Sciences 15, 2007, 31-36.
We present some elementary integrals that are combinations of logarithms and exponentials.
- The integrals in Gradshteyn and Ryzhik. Part 2: elementary logarithmic integrals
Scientia, Series A: Math. Sciences 14, 2007, 7-15.
We present the evaluation of elementary integrals of the form
R(x) times log(x). The results involve the dilogarithm function.
- The integrals in Gradshteyn and Ryzhik. Part 1: a family of logarithmic integrals
Scientia, Series A: Math. Sciences 14, 2007, 1-6.
We present the evaluation of a family of integrals of the form: power of log x times
a rational function with two poles.
- Combinatorial sequences arising from a rational integral ( 15 pages)
Online Journal of Analytic Combinatorics Issue 2 (2007), #4.
The 2-adic valuation of a sequence of coefficients that have appeared in the evaluation
of a rational integral is given a combinatorial interpretation.
- Rational Landen transformations on the real line (with Dante Manna) ( 22 pages)
Math Comp 76, 2007, 2023-2043.
The classical elliptic Landen transformations are extended to the rational
case. Explicit
formulas are provided.
- A simple example of a new class of Landen transformations (with Dante Manna) (12 pages)
American Mathematical Monthly March 2007, 114, 232-241.
The rational Landen transformation is illustrated in the case of a quadratic rational function.
- Dynamics of the degree six Landen transformation (with Marc Chamberland) (15 pages)
Discrete and Continuous Dynamical Systems 15, 2006, 905-919.
We prove in a dynamical manner that the degree six Landen transformation converges precisely
when the integral that generates it is finite.
- A summation method due to Carr: part 1
(with G. Boros)
Scientia, Series A: Math. Sciences 12, 2006, 21-37.
(17 pages). We use a simple technique that appears in the classical
book of Carr, accesible to Ramanujan, to evaluate several integrals.
- The evaluation of Tornheim double sums, Part I (with Olivier Espinosa) (30 pages)
Journal of Number Theory 116, 2006, 200-229.
Explicit formula for Tornheim double series are given in terms of integrals
involving
the Hurwitz zeta function.
- Sums of arctangents and some formulas of Ramanujan (with G. Boros) (12 pages)
Scientia, Series A: Math. Sciences 11, 2005, 13-24.
We present some analytic evaluation of arctangent sums.
- An elementary evaluation of a quartic integral (with G. Boros and S. Riley) (12 pages)
Scientia, Series A: Math. Sciences 11, 2005, 1-12.
We present an elementary discussion of the evaluation of a quartic
rational integral.
- A map on the space of rational functions (with G. Boros, J. Little, E. Mosteig and R. Stanley) (20 pages)
Rocky Mountain Journal 35, 2005, 1861-1880.
We describe a map on the space of rational functions. We classify all its fixed points.
- A generalized polygamma function (with Olivier Espinosa) (15 pages)
Integral Transforms and Special Functions 15, 2004, 101-115.
We introduce a function of two complex variables that generalizes the
polygamma and
negapolygamma functions. Some integrals of this function are
explicitly evaluated.
- On some families of integrals solvable in terms of polygamma and negapolygamma functions (with
George Boros and Olivier Espinosa) (17 pages)
Integral Transforms and Special Functions 14, 2003, 187-203.
We introduce negapolygamma functions and use it to evaluate some definite integrals.
- A geometric view of the rational Landen transformation (with John Hubbard) (9 pages)
Bull. London Math. Soc. 35, 2003, 293-301.
The rational Landen transformation is interpreted as the direct image of a rational function
by the map w(z) = (z^2-1)/2z. Convergence of its iterates is established.
- The evaluation of integrals: a personal story (7 pages)
Notices Amer. Math. Soc. 49, March 2002, 311-317.
A description of how I got involved in the evaluation of integrals.
- A transformation of rational functions (with G. Boros and M. Joyce) (11 pages)
Elemente der Mathematik 57, 2002, 1-11.
We describe a map on the space of rational functions. Examples of fixed points and periodic orbits are provided.
- Bernoulli on arc length (with J. Nowalsky, G. Roa and L. Solanilla) (5 pages)
Math. Magazine 75, 2002, 209-213
We examine Bernoulli's original paper and his contributions to the evaluation of integrals dealing
with length of curves.
- On some definite integrals involving the Hurwitz zeta function. Part 2. (with Olivier Espinosa) (20 pages)
Ramanujan Journal 6, 2002, 449-468.
We produce closed form evaluations of several definite and indefinite integrals involving the Hurwitz zeta function.
- On some definite integrals involving the Hurwitz zeta function. Part 1. (with Olivier Espinosa) (30 pages)
Ramanujan Journal 6, 2002, 159-188.
We produce closed form evaluations of several definite integrals involving the Hurwitz zeta function.
- Landen transformations and the integration of rational functions. (with George Boros) (20 pages)
Math Comp 71, 2002, 649-668.
We present a generalization of the classical Landen transformation to the class of even rational functions.
The algorithm
takes the coefficients of an even rational function and produces a new function of the same type,
with the same integral.
Numerical evidence of convergence is presented.
- An integral with three parameters. Part 2. (with George Boros and Roopa Nalam) (14 pages)
Jour. Comp. Appl. Math. 134, 2001, 113-126.
We discuss more examples of definite integrals that can be evaluated from a
single integral with three free parameters.
- An extension of a criterion for unimodality (with J. Alvarez, M. Amadis, G. Boros, D. Karp and L. Rosales) (6 pages)
Elec. Jour. Combinatorics 8, 2001, #R30.
We show that if P(x) is a polynomial with nondecreasing, nonnegative coefficients, then the
coefficient sequence of P(x+n) is unimodal for any natural n.
This paper was a result obtained in SIMU 2000.
- The double square root, Jacobi polynomials and Ramanujan's Master Theorem (with George Boros) (9 pages)
Jour. Comp. Appl. Math. 130, 2001, 337-344.
We show that the polynomials P(a;m) that appeared in our work on the quartic integral, also
appear in the expansion of the double square root function. The
polynomials P(a;m) are
then identified as part of the Jacobi family.
- A property of Euler's elastic curve. (with P. Neill, J. Nowalsky and L. Solanilla) (7 pages)
Elemente der Mathematik 55, 2000, 1-7.
We study some integrals that appeared in Euler's work on elastic curves.
- A rational Landen transformation. The case of degree six. (with George Boros) (9 pages)
Contemporary Mathematics 251, 2000, 83-91.
The classical Landen transformation of elliptic integrals is extended to rational integrands. The case
of degree six is given in detail.
The paper is reviewed in here
- The 2-adic valuation of the coefficients of a polynomial (with George Boros and Jeffrey Shallit) (14 pages)
Scientia, Series A: Math. Sciences 7, 2000, 47-60. Special issue in the memory of Miguel Blazquez.
We establish an analytic expression for the 2-adic valuation of the linear coefficient of a polynomial
that appeared in the evaluation of a quartic integral.
- The integration of rational functions:
examples and problems (with George Boros) (20 pages)
Scientia, Series A: Math. Sciences 6, 2000, 9-28.
We present some new algorithms and problems related to the evaluation
of definite integrals.
- A criterion for unimodality (with George Boros) (4 pages)
Elec. Jour. Combinatorics 6, 1999, #R10.
We show that if P(x) is a polynomial with nondecreasing, nonnegative coefficients, then the
coefficient sequence of P(x+1) is unimodal.
- A sequence of unimodal polynomials (with George Boros) (15 pages)
Journal of Math. Anal. and Applications 237, 1999, 272-287.
The unimodality of a sequence of coefficients arising from the evaluation of a quartic integrals is
shown to be unimodal. We conjecture that these coefficients are logconcave. The zeros of the
corresponding polynomials are studied and we conjecture an expression for their limiting behavior.
- An integral hidden in Gradshteyn and Ryzhik (with George Boros) (7 pages)
Jour. Comp. Appl. Math. 106, 1999, 361-368.
The integral of the power of a rational function is
shown to be hidden in a classical table of integrals.
February 2009
Update.
- An integral with three parameters (with George Boros) (9 pages)
SIAM Review 40, 1998, 972-980
An integral with three free parameters is evaluated. Many classical evaluations are presented as
special cases.
The paper is reviewed in here
January 2009
Update.
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