Papers by Victor H. Moll
If you would like a hard copy, please email me at vhm at math.tulane.edu
-
Asymptotics and exact formulas for Zagier polynomials
(with A. Dixit, M. L. Glasser, C. Vignat) (27 pages)
Res. Number Theory 12 (2016) 47-74.
Expressions for the Zagier polynomials, reminiscent of the classical
Fourier expansions of Bernoulli polynomials are presented. These are used
to derive asymptotic formulas for these polynomials.
-
Periodicity in the p-adic valuation of a polynomial
(with L. Medina and E. Rowland) (16 pages)
Journal of Number Theory 180 (2017) 139-153.
The p-adic valuation of an integer is the largest power of the prime p that
divides it. This work describes the p-adic valuation of polynomials.
This set is either periodic or unbounded. The period is given in the
first case.
- A series involving Catalan numbers. Proofs and demonstrations
(with T.Amdeberhan, X.Guan, L.Jiu and C.Vignat) (12 pages)
Elemente der Mathematik 71 (2016) 109-121.
The formula for the generating function of the reciprocal of Catalan numbers
is established by a variety of methods.
-
A closed-form solution might be given by a tree. Valuations of quadratic polynomials
(with A. Byrnes, J. Fink, G. Lavigne, I. Nogues, S. Rajasekaran,
A. Yuan, L. Almodovar, X. Guan,
A. Kesarwani, L. Medina and E. Rowland) (17 pages)
To appear in ??
The p-adic valuation of an integer is the largest power of the prime p that
divides it. This work describes the 2-adic valuation of quadratic polynomials.
- A symbolic approach to some Bernoulli-Barnes polynomials
(with L.Jiu and C.Vignat) (12 pages)
International Journal of Number Theory 12 (2016) 109-121.
A symbolic method is used to establish some properties of the
Bernoulli-Barnes polynomials. The results are motivated by recent work of
A. Bayad and M. Beck.
- A symbolic approach to multiple zeta values at the negative integers
(with L.Jiu and C.Vignat)
Journal of Symbolic Computation
84 (2018) 1-13.
The analytic continuation of MZV values is described by symbolic methods.
- The moments of the hydrogen atom by the method of brackets
(with I. Gonzalez, K. Kohl, I. Kondrashuk and D. Salinas) (13 pages)
SIGMA 13 (2017) 1-13.
The method of brackets is used to evaluate a family of integrals that
give the expected value of power of the radial coordinate in arbitrary
hydrogen states in the quantum case.
- Modified Norlund polynomials
(with A. Dixit, A. Kabza and C. Vignat) (22 pages)
The Ramanujan Journal
42 (2017) 69-96.
The Norlund polynomials are defined by the generating function that is a
power of the corresponding generating function for the Bernoulli numbers.
These polynomials are modified as Zagier did for the classical case. This
paper contains an expression for the generating function of the modfied case.
- Involutions and their progenies
(with T. Amdeberhan)
Journal of Combinatorics
6 (2016) 483-508.
Combinatorial and arithmetic properties of the sequence counting involutions
and other related sequences are presented.
- Self-reciprocal functions, powers of the Riemann zeta function and modular-type transformations
(with A. Dixit) (34 pages)
Journal of Number Theory 147 (2015) 211-249.
The classical transformation of Jacobi's theta function admits a simple proof
by producing an integral representation that yields this invariance apparent.
This idea seems to have first
appeared in the work of S. Ramanujan.
Several examples of this idea have been produced by Koshlyakov,
Ferrar,
Guinand, Ramanujan and others. A unifying procedure to analyze these examples
and natural
generalizations is presented.
- A hypergeometric inequality
(with A. Dixit and V. Pillwein) (7 pages)
Journal of Analytic Combinatorics
20 (2016) 65-72.
An inequality for hypergeometric functions, encounted in a proof
of unimodality, is given an automatic proof.
- The finite Fourier transform of classical polynomials
(with Atul Dixit, Lin Jiu and Christophe Vignat) (16 pages)
Journal of the Australian Mathematical Society 98 (2015) 145-160.
The finite Fourier transform of a family of orthogonal polynomials is the
usual transform of the function extended as zero outside its natural domain.
Explicit formulas for the Fourier transform of Legendre, Jacobi, Gegenbauer and
Chebyshev polynomials are presented.
- Generalized Bernoulli numbers and a formula of Lucas
(with Christophe Vignat) (11 pages)
The Fibonacci Quarterly
53 (2015) 349-359.
The generalized Bernoulli numbers come from the generating function
(z/(exp(z)-1)) to the p-the power.
An interesting, not very well-known, formula due to E. Lucas is given a new proof.
An expression in terms of the Meixner-Pollaczek polynomials is presented.
- Identities for generalized Euler polynomials
(with Lin Jiu and Christophe Vignat) (12 pages)
Integral Transforms and Special Functions 25 (2014)
777-789.
A generalization of the classical Euler polynomials is discussed.
A random variable defined by Klebanov in terms
in terms of the Chebyshev polynomials is used to express the classical Euler polynomials in terms of the
generalized ones. Asymptotics of these probability numbers are established.
- Generalized Fibonacci polynomials and Fibonomial coefficients
(with T. Amdeberhan, X. Chen and B. Sagan) (19 pages)
Annals of Combinatorics 18 (2014) 541-562.
A family of polynomials generalizing Fibonacci numbers is introduced.
Combinatorial interpretations are provided.
- On polynomials connected to powers of Bessel functions
(with C. Vignat) (11 pages)
International Journal of Number Theory 10 (2014) 1245-1257.
A family of polynomials that appears in the expansion of powers of Bessel
function
is investigated. Two kinds of recurrences are established. Connections with
the umbral
formalism of Cholewinski are presented.
- Recursion rules for the hypergeometric zeta function
(with A. Byrnes, Lin Jiu and C. Vignat) (19 pages)
International Journal of Number Theory 7 (2014) 1761-1782.
The hypergeometric zeta function is defined in terms of the zeros of the Kummer function.
Some of its analytic and arithmetic properties are discussed.
In particular we establish two
linear recurrences and one quadratic one.
- The unimodality of a polynomial coming from a rational integral. Back to the original proof
(with T. Amdeberhan, A. Dixit, Xiao Guan and Lin Jiu) (12 pages)
Journal of Math. Anal. and Applications 420, 2014, 1154-1166.
A sequence of coefficients is shown to be unimodal. The proof presented here
is based on a
monotonicity argument. It is an improved hypergeometric approach to
the original proof.
- From sequences to polynomials and back, via operator orderings
(with T. Amdeberhan, V. De Angelis, A. Dixit, and C. Vignat) (16 pages)
Journal of Mathematical Physics 54 (2013) 123502.
C. M. Bender and G. V. Dunne showed that linear combinations
of words $q^{k}p^{n}q^{n-k}$ where
p and q are subject to the relation qp − pq = ı,
may be expressed as a polynomial in the symbol z =
(qp + pq)/2.
Relations
between such polynomials and linear combinations of the transformed
coefficients are explored.
In particular, examples yielding orthogonal polynomials
are provided.
- A special rational function with vanishing integral (5 pages)
Integral Transforms and Special Functions 24 (2014) 970-975.
The evaluation of a quartic integral that recently appeared in Mathematics
Stack Exchange is generalized.
- The Zagier polynomials. Part II: arithmetic properties of coefficients.
(with M. Coffey, V. De Angelis, A. Dixit, A. Straub and C. Vignat) (25 pages)
The Ramanujan Journal
35 (2014) 361-390.
In 1998 Zagier presented a variation of the Bernoulli numbers and
established that certain subsequences had
period 6. An extension of this variation obtained by replacing Bernoulli numbers by Bernoulli polynomials was
recently proposed by some of the authors. This paper examines the divisibility properties of the coefficients.
For instance, it is shown that the odd powers have coefficients with denominator exactly 4.
- The Zagier modification of Bernoulli numbers and a polynomial extension. Part I.
(with A. Dixit and C. Vignat) (35 pages)
The Ramanujan Journal 33, 2014, 379-422.
In 1998 Zagier presented a variation of the Bernoulli numbers and
established that certain subsequences had
period 6. An extension of this variation is obtained by replacing Bernoulli numbers by Bernoulli polynomials.
A complete classification of values that give periodic subsequences is presentes. There is a nice connection
with Chebyshev polynomials (of both kinds). The proof empoys the umbral method.
- An arithmetic conjecture on a sequence of arctangent sums
(27 pages)
Scientia,
Series A: Math. Sciences 24, 2013, 90-119.
A nonlinear sequence, coming from arctangent sums, has been conjectured not to
take integer values. This problem is studied in terms of an auxiliary sequence
with interesting arithmetic properties.
- Valuations and combinatorics of truncated exponential sums
(with T. Amdeberhan and D. Callan) (15 pages)
INTEGERS,
Volume 13, paper A21, 2013.
Arithmetic properties of the n-th partial sum of e^n are determined.
The case p=2 solves a conjecture of G. McGarvey. Some combinatorial
interpretations for these partial sums are provided in terms of rooted trees.
- A probabilistic interpretation of a sequence related to Narayana polynomials
(with T. Amdeberhan and C. Vignat) (26 pages)
Online Journal of Analytic Combinatorics 8, Paper 3, 2013.
A sequence appearing in a recurrence for Narayana polynomials is generalized
and is expressed in terms of random variables. Arithmetical properties of this
generalized sequence are investigated.
- A probabilistic approach to some binomial identities
(with C. Vignat) (11 pages)
Elemente der Mathematik 69, 2014, 1-12.
Some identiities involving binomial coefficients are given probabilistic proofs.
- A family of palindromic polynomials
(with T. Amdeberhan and V. Papanicolau
(7 pages)
Scientia,
Series A: Math. Sciences 24, 2013, 25-32.
The zeros of a family of polynomials are described.
- Complementary Bell numbers: Arithmetical properties and Wilf's conjecture
(with T. Amdeberhan and V. De Angelis)
(27 pages)
Analytic expression for the $2$-adic valuation of Bell and complementary Bell numbers
are provided. Wilf conjectured that the only vanishing complementary Bell
numbers is the third one.
It is shown that there is at most more example.
- Ramanujan Master Theorem applied to the evaluation fo Feynman diagrams
(with I. Gonzalez and I. Schmidt)
(11 pages)
A multi-dimensional version of Ramanujan Master Theorem is
employed to evaluate integrals
coming from Feynman diagrams.
- Ramanujan Master's Theorem
(with T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Harrison and A. Straub)
(20 pages)
The Ramanujan Journal 29, 2012, 103-120.
One of Ramanujan's favorite technique to evaluate definite integrals is
surveyed.
Applications to integrals of classical polynomials, random walk
integrals and
examples coming from the evaluation of Feynman diagrams
illustrate this powerful method.
- The p-adic valuation of ASM numbers.
(with Erin Beyerstedt and Xinyu Sun) (11 pages)
Journal of Integer Sequences 14, 2011, article 11.8.7
We provide a formula, similar to Legendre's series for factorials, for the
p-adic valuation
of the ASM numbers. These numbers count size n
matrices formed
by 0, +1 and -1 with
column and row sum equal to 1 and alternating non-zero entries.
- The iterated integrals of ln(1+x^2)
(with Tewodros Amdeberhan, Christoph Koutschan and Eric Rowland)
(20 pages).
International Journal of Number Theory 7, 2011, 1-12.
Iterating the primitives of ln(1+x^2) produces a sequence
of polynomials. Some of its properties
are described.
- Broken bracelets, Molien series, paraffin wax
and an elliptic curve of conductor 48
(with T. Amdeberhan and Mahir Can) ( 19 pages).
Certain enumeration questions arising from the study of bracelets are solved.
Applications and interpretations are provided.
SIAM Journal of Discrete Math. 25, 2011,1843-1859.
- A by-product of an integral evaluation
(with T. Amdeberhan) (4 pages).
M. Apagodu used the Almkvist-Zeilberger method to evaluate a quartic
integral. A new proof, using Schur functions, is provided.
To appear in The Ramanujan Journal
- Arithmetic properties of plane partitions
(with T. Amdeberhan) (6 pages).
The 2-adic valuations of the number of alternating sign
matrices of size n and the number of
totally symmetric plane partitions
are shown to be related in a simple manner.
Elec. Jour. Combinatorics 18, 2011, #R1.
- Solution to Problem 11519
(with T. Amdeberhan) (2 pages).
We present a solution to Problem 11519 by Ovidiu Furdui.
- The evaluation of a quartic integral via Schwinger, Schur and Bessel
(with T. Amdeberhan and Christophe Vignat) (14 pages).
The evaluation of a quartic integral is given by three different methods.
The first one involves a formula of Schwinger for integrals coming from
Feynman diagrams. The second one employs Schur functions and the third one
uses a representation in terms of Bessel functions.
The Ramanujan Journal 28, 2012,1-14.
- A pretty binomial identity
(with T. Amdeberhan, V. de Angelis, M. Lin and B. Sury) (6 pages).
An identity involving binomial coefficients that appeared in the
evaluation of a quartic integral is established by a variety of methods.
Elemente der Mathematik 67, 2012, 1-8.
- Integrals of powers of loggamma
(with Tewodros Amdeberhan, Mark W. Coffey, Olivier Espinosa,
Christoph Koutschan and Dante V. Manna)
(11 pages).
Proc. Amer. Math. Soc.
139, 2011, 535-545.
The integrals of powers of LogGamma form 0 to 1 are considered. The first
two powers have
analytic expressions.
The third power is still an open question. Our attempt to produce such an
expression is discussed.
- The action of Hecke operators on hypergeometric functions
Journal of Australian Mathematical Society 89, 2010, 51-74
(with Sinai Robins and Kirk Soodhalter) (22 pages).
We study the spectrum of Hecke operators acting on hypergeometric functions.
Polylogarithms appear as eigenfunctions.
- The method of brackets. Part 2: examples and applications (with Ivan Gonzalez and Armin Straub)
Contemporary Mathematics 517, 157-171, 2010.
The method of brackets is a heuristic algorithm for integration. Examples are
discussed.
- Iterated primitives of logarithmic
powers (with Luis Medina and Eric Rowland)
(12 pages).
International Journal of Number Theory 7(3), 623-634, 2011
Iterating the primitives of powers of logarithms produces a sequence
of polynomials. These are shown to be
logconcave and the arithmetical properties
of its coefficients are presented.
- The $p$-adic valuation of Stirling
numbers (with A. Berrizbeitia, L. Medina, A. Moll and L. Noble)
Journal for Algebra and Number Theory Academia 1, 2010, 1-30.
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).
American Mathematical Monthly 117, 2010, 618-632
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 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.
- 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.
Numerische Mathematik115, 2010, 289-307.
- Seized opportunities. (14 pages)
We describe some of the Mathematical issues involved in the
evaluation of definite integrals.
Notices Amer. Math. Soc.
57, 2010, 476-484.
- The evaluation of Tornheim double sums, Part 2 (with Olivier Espinosa)
The Ramanujan Journal 22, 2010, 55-99.
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.
- Definite integrals by the method of brackets. Part 1. (with Ivan Gonzalez)
Advances in Applied Mathematics 45, 2010, 50-73.
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.
- 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.
Journal of Math. Physics 50, 2009, 103528
- The p-adic valuations of sequences counting alternating sign matrices. (with Xinyu 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
- 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 $p$-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.
Acta Arithmetica 149, 2009, 31-42.
- A binary tree representation for the 2-adic valuation of a sequence arising from a rational integral
(with Xinyu Sun) (10 pages).
Integers 10, 2009, 211-222.
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.
- 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.
- 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.
- 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 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 $2$-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.
- 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.
- The story of Landen, the hyperbola and the ellipse. (with J. Nowalsky and L. Solanilla) (7 pages)
Elemente der Mathematik 57, 2002, 19-25.
We describe integrals appearing in Landen's work on conics.
- 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, 156-162.
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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