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BEGIN:VEVENT
SUMMARY:Arvind Ayyer (IISc\, Bangalore)
DTSTART:20200618T103000Z
DTEND:20200618T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/1
DESCRIPTION:Title: The Monopole-Dimer Model\nby Arvind Ayyer (IISc\, Bangalore)
as part of Special Functions and Number Theory seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hjalmar Rosengren (Chalmers University of Technology and the Unive
rsity of Gothenburg)
DTSTART:20200702T092500Z
DTEND:20200702T103000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/2
DESCRIPTION:Title: On the Kanade-Russell identities\nby Hjalmar Rosengren (Chalm
ers University of Technology and the University of Gothenburg) as part of
Special Functions and Number Theory seminar\n\n\nAbstract\nKanade and Russ
ell conjectured several Rogers-Ramanujan-type identities for triple series
. Some of these conjectures are related to characters of affine Lie algebr
as\, and they can all be interpreted combinatorially in terms of partition
s. Many of these conjectures were settled by Bringmann\, Jennings-Shaffer
and Mahlburg. We describe a new approach to the Kanade-Russell identities\
, which leads to new proofs of five previously known identities\, as well
as four identities that were still open. For the new cases\, we need quadr
atic transformations for q-orthogonal polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ali Uncu (RICAM\, Austrian Academy of Sciences)
DTSTART:20200716T093000Z
DTEND:20200716T103000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/3
DESCRIPTION:Title: The Mathematica package qFunctions for q-series and partition the
ory applications\nby Ali Uncu (RICAM\, Austrian Academy of Sciences) a
s part of Special Functions and Number Theory seminar\n\n\nAbstract\nIn th
is talk\, I will demonstrate the new Mathematica package qFunctions while
providing relevant mathematical context. This implementation has symbolic
tools to automate some tedious and error-prone calculations and it also in
cludes some other functionality for experimentation. We plan to highlight
the four main tool-sets included in the qFunctions package:\n\n(1) The q-d
ifference equation (or recurrence) guesser and some formal manipulation to
ols\,\n(2) the treatment of the method of weighted words and automatically
finding and uncoupling recurrences\,\n\n(3) a method on the cylindrical p
artitions to establish sum-product identities\,\n(4) fitting polynomials w
ith suggested well-known objects to guess closed formulas.\n\nThis talk is
based on joint work with Jakob Ablinger (RISC).\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alok Shukla (Ahmedabad University)
DTSTART:20200730T102500Z
DTEND:20200730T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/4
DESCRIPTION:Title: Tiling proofs of Jacobi triple product and Rogers-Ramanujan ident
ities\nby Alok Shukla (Ahmedabad University) as part of Special Functi
ons and Number Theory seminar\n\n\nAbstract\nThe Jacobi triple product ide
ntity and Rogers-Ramanujan identities are among the most famous q-series i
dentities. We will present elementary combinatorial "tiling proofs" of the
se results. The talk should be accessible to a general mathematical audien
ce.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fatma Cicek (IIT\, Gandhinagar)
DTSTART:20200813T103000Z
DTEND:20200813T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/5
DESCRIPTION:Title: On the logarithm of the Riemann zeta-function near the nontrivial
zeros\nby Fatma Cicek (IIT\, Gandhinagar) as part of Special Function
s and Number Theory seminar\n\n\nAbstract\nSelberg's central limit theorem
is one of the most significant probabilistic results in analytic number t
heory. Roughly\, it states that the logarithm of the Riemann zeta-function
on and near the critical line has an approximate two-dimensional Gaussian
distribution.\n\nIn this talk\, we will talk about our recent result whic
h states that the distribution of the logarithm of the Riemann zeta-functi
on near the sequence of the nontrivial zeros has a similar central limit t
heorem. Our results are conditional on the Riemann Hypothesis and/or suita
ble zero-spacing hypotheses. They also have suitable generalizations to Di
richlet $L$-functions.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amritanshu Prasad (IMSc\, Chennai)
DTSTART:20200827T103000Z
DTEND:20200827T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/6
DESCRIPTION:Title: Character Polynomials and their Moments\nby Amritanshu Prasad
(IMSc\, Chennai) as part of Special Functions and Number Theory seminar\n
\n\nAbstract\nA polynomial in a sequence of variables can be regarded as a
class \nfunction on every symmetric group when the $i$th variable is inte
rpreted as \nthe number of $i$-cycles. Many nice families of representatio
ns of symmetric \ngroups have characters represented by such polynomials.\
n\nWe introduce two families linear functionals of this space of polynomia
ls -- moments and signed moments. For each $n$\, the moment of a polynomia
l at $n$ \ngives the average value of the corresponding class function on
the $n$th \nsymmetric group\, while the signed moment gives the average of
its \nproduct by the sign character. These linear functionals are easy to
\ncompute in terms of binomial bases of the space of polynomials.\n\nWe u
se them to explore some questions in the representation theory of \nsymmet
ric groups and general linear groups. These explorations lead to \ninteres
ting expressions involving multipartite partition functions and \nsome pec
uliar variants.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (IISc.\, Bangalore)
DTSTART:20200910T103000Z
DTEND:20200910T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/7
DESCRIPTION:Title: An introduction to total positivity\nby Apoorva Khare (IISc.\
, Bangalore) as part of Special Functions and Number Theory seminar\n\n\nA
bstract\nI will give a gentle introduction to total positivity and the the
ory of Polya frequency (PF) functions. This includes their spectral proper
ties\, basic examples including via convolution\, and a few proofs to show
how the main ingredients fit together. Many classical results (and one Hy
pothesis) from before 1955 feature in this journey. I will end by describi
ng how PF functions connect to the Laguerre-Polya class and hence Polya-Sc
hur multipliers\, and mention 21st century incarnations of the latter.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (IISc.\, Bangalore)
DTSTART:20200924T103000Z
DTEND:20200924T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/8
DESCRIPTION:Title: Totally positive matrices\, Polya frequency sequences\, and Schur
polynomials\nby Apoorva Khare (IISc.\, Bangalore) as part of Special
Functions and Number Theory seminar\n\n\nAbstract\nI will discuss totally
positive/non-negative matrices and kernels\, including Polya frequency (PF
) functions and sequences. This includes examples\, history\, and basic re
sults on total positivity\, variation diminution\, sign non-reversal\, and
generating functions of PF sequences (with some proofs). I will end with
applications of total positivity to old and new phenomena involving Schur
polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debashis Ghoshal (School of Physical Sciences\, JNU)
DTSTART:20201008T103000Z
DTEND:20201008T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/9
DESCRIPTION:Title: Two-dimensional gauge theories\, intersection numbers and special
functions\nby Debashis Ghoshal (School of Physical Sciences\, JNU) as
part of Special Functions and Number Theory seminar\n\n\nAbstract\nThe pa
rtition function of two dimensional Yang-Mills theory contains a wealth of
information about the moduli space of connections on surfaces. We study t
his problem on a special class of surfaces of infinite genus\, which are c
onstructed recursively. While the results are suggestive of an underlying
geometrical structure\, we use it as a prop to efficiently compute results
for finite genus surfaces. Riemann zeta function\, confluent hypergeometr
ic function and its truncations show up in explicit computations for the g
auge group SU(2). Much of the corresponding results are open for other gro
ups.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Schlosser (Universitat Wien)
DTSTART:20201022T103000Z
DTEND:20201022T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/10
DESCRIPTION:Title: Basic hypergeometric proofs of two quadruple equidistributions o
f Euler--Stirling statistics on ascent sequences\nby Michael Schlosser
(Universitat Wien) as part of Special Functions and Number Theory seminar
\n\n\nAbstract\nIn my talk\, I will present new applications of basic\nhyp
ergeometric series to specific problems in enumerative\ncombinatorics. The
combinatorial problems we are interested in\nconcern multiply refined equ
idistributions on ascent sequences.\n(I will gently explain these notions
in my talk!)\nUsing bijections we are able to suitably decompose some\nqua
druple distributions we are interested in and obtain\nfunctional equations
and ultimately generating functions\nfrom them\, in the form of explicit
basic hypergeometric series\,\nThe problem of proving equidistributions th
en reduces to\napplying suitable transformations of basic hypergeometric s
eries.\nThe situation in our case however is tricky (caused by the\nfact h
ow the power series variable $r$ appears in the base\n$q=1-r$ of the respe
ctive basic hypergeometric series\; so being\ninterested in the generating
function in $r$ as a Maclaurin\nseries\, we are thus interested in the an
alytic expansion of\nthe nonterminating basic hypergeometric series in bas
e $q$\naround the point $q=1$)\, as none of the known transformations\napp
ear to directly work to settle our problems\; we require\nthe derivation o
f new identities.\nSpecifically\, we use the classical Sears transformatio
n and\napply some analytic tools to establish a new non-terminating\n${}_4
\\phi_3$ transformation formula of base $q$\, valid as\nan identity in a n
eighborhood around $q=1$. We use special\ncases of this formula to deduce
two different quadruple\nequidistribution results involving Euler--Stirlin
g statistics\non ascent sequences. One of them concerns a symmetric\nequi
distribution\, the other confirms a bi-symmetric\nequidistribution that wa
s recently conjectured in a paper\n(published in JCTA) by Shishuo Fu\, Emm
a Yu Jin\, Zhicong Lin\,\nSherry H.F. Yan\, and Robin D.B. Zhou. This is
joint work\nwith Emma Yu Jin. For full results (and further ones)\, see\nh
ttps://arxiv.org/abs/2010.01435\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanoli Gun (IMSc\, Chennai)
DTSTART:20201112T103000Z
DTEND:20201112T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/11
DESCRIPTION:Title: Large values of $L$-functions\nby Sanoli Gun (IMSc\, Chennai
) as part of Special Functions and Number Theory seminar\n\n\nAbstract\nIn
this lecture\, we will give an overview of a method of Soundararajan and
show that how this method can be used to produce large values of $L$-funct
ions in different set-ups.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manjil P. Saikia (Cardiff University)
DTSTART:20201126T103000Z
DTEND:20201126T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/12
DESCRIPTION:Title: Refined enumeration of symmetry classes of Alternating Sign Matr
ices\nby Manjil P. Saikia (Cardiff University) as part of Special Func
tions and Number Theory seminar\n\n\nAbstract\nThe sequence $1\,1\,2\,7\,4
2\,429\, \\ldots$ counts several combinatorial objects\, some of which I w
ill describe in this talk. The major focus would be one of these objects\,
alternating sign matrices (ASMs). ASMs are square matrices with entries i
n the set {0\,1\,-1}\, where non-zero entries alternate in sign along rows
and columns\, with all row and column sums being 1. I will discuss some q
uestions that are central to the theme of ASMs\, mainly dealing with their
enumeration. In particular we shall prove some conjectures of Fischer\, R
obbins\, Duchon and Stroganov. This talk is based on joint work with Ilse
Fischer and some ongoing work.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sneha Chaubey (IIIT\, Delhi)
DTSTART:20201210T103000Z
DTEND:20201210T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/13
DESCRIPTION:Title: Generalized visible subsets of two dimensional integer lattice\nby Sneha Chaubey (IIIT\, Delhi) as part of Special Functions and Numbe
r Theory seminar\n\n\nAbstract\nWe will discuss some subsets of two-dimens
ional integer lattice which arise as visible sets under some suitable noti
on of visibility. We will discuss some set-theoretic (Delone\, Meyer\, Qua
sicrystals etc.)\, geometrical (density and gaps) and dynamical (auto-corr
elation and diffraction pattern) properties of these subsets\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rahul Kumar (IIT\, Gandhinagar)
DTSTART:20201217T103000Z
DTEND:20201217T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/14
DESCRIPTION:Title: A generalized modified Bessel function and explicit transformati
ons of certain Lambert series\nby Rahul Kumar (IIT\, Gandhinagar) as p
art of Special Functions and Number Theory seminar\n\n\nAbstract\nAn exact
transformation\, which we call a master identity\, is obtained for the se
ries $\\sum_{n=1}^{\\infty}\\sigma_{a}(n)e^{-ny}$ for $a\\in\\mathbb{C}$ a
nd Re$(y)>0$. As corollaries when $a$ is an odd integer\, we derive the we
ll-known transformations of the Eisenstein series on $\\textup{SL}_{2}\\le
ft(\\mathbb{Z}\\right)$\, that of the Dedekind eta function as well as Ram
anujan's famous formula for $\\zeta(2m+1)$. Corresponding new transformati
ons when $a$ is a non-zero even integer are also obtained as special cases
of the master identity. These include a novel companion to Ramanujan's fo
rmula for $\\zeta(2m+1)$. Although not modular\, it is surprising that suc
h explicit transformations exist. The Wigert-Bellman identity arising from
the $a=0$ case of the master identity is derived too. The latter identity
itself is derived using Guinand's version of the Vorono\\"{\\i} summation
formula and an integral evaluation of N. S. Koshliakov involving a genera
lization of the modified Bessel function $K_{\\nu}(z)$. Koshliakov's integ
ral evaluation is proved for the first time. It is then generalized using
a well-known kernel of Watson to obtain an interesting two-variable genera
lization of the modified Bessel function. This generalization allows us to
obtain a new transformation involving the sums-of-squares function $r_k(n
)$. This is joint work with Atul Dixit and Aashita Kesarwani.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wadim Zudilin (Radboud University)
DTSTART:20210107T103000Z
DTEND:20210107T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/15
DESCRIPTION:Title: Ramanujan special talk: 10 years of q-rious positivity. More nee
ded!\nby Wadim Zudilin (Radboud University) as part of Special Functio
ns and Number Theory seminar\n\n\nAbstract\nThe $q$-binomial coefficients
\\[ \\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)\,\\] for integers $0\\le m\\le n$\
, are known to be polynomials with non-negative integer coefficients. This
readily follows from the $q$-binomial theorem\, or the many combinatorial
interpretations of them. Ten years ago\, together with Ole Warnaar we obs
erved that this non-negativity (aka positivity) property generalises to pr
oducts of ratios of $q$-factorials that happen to be polynomials\; we prov
e this observation for (very few) cases. During the last decade a resumed
interest in study of generalised integer-valued factorial ratios\, in conn
ection with problems in analytic number theory and combinatorics\, has bro
ught to life new positive structures for their $q$-analogues. In my talk I
will report on this "$q$-rious positivity" phenomenon\, an ongoing projec
t with Warnaar.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josef Kustner (University of Vienna)
DTSTART:20210121T103000Z
DTEND:20210121T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/16
DESCRIPTION:Title: Elliptic and $q$-analogs of the Fibonomial numbers\nby Josef
Kustner (University of Vienna) as part of Special Functions and Number Th
eory seminar\n\n\nAbstract\nThe Fibonomial numbers are integer numbers obt
ained from the binomial coefficients by replacing each term by its corresp
onding Fibonacci number. In 2009\, Sagan and Savage introduced a simple co
mbinatorial model for the Fibonomial numbers. \n\nIn this talk\, I will pr
esent a combinatorial description for a q-analog and an elliptic analog of
the Fibonomial numbers which is achieved by introducing certain q- and el
liptic weights to the model of Sagan and Savage.\n\nThis is joint work wit
h Nantel Bergeron and Cesar Ceballos.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Moll (Tulane University)
DTSTART:20210204T103000Z
DTEND:20210204T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/17
DESCRIPTION:Title: Valuations of interesting sequences\nby Victor Moll (Tulane
University) as part of Special Functions and Number Theory seminar\n\n\nAb
stract\nGiven a sequence ${ a_{n} }$ of integers and a prime $p$\, the seq
uence of\nvaluation $nu_{p}(a_{n})$ presents interesting challenges. This
talk will discuss a\nvariety of examples in order to illustrate these cha
llenges and present our approach\nto this problem.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gaurav Bhatnagar (Ashoka University)
DTSTART:20210218T103000Z
DTEND:20210218T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/18
DESCRIPTION:Title: The Partition-Frequency Enumeration Matrix\nby Gaurav Bhatna
gar (Ashoka University) as part of Special Functions and Number Theory sem
inar\n\n\nAbstract\nWe develop a calculus that gives an elementary approac
h to enumerate partition-like objects using an infinite upper-triangular n
umber-theoretic matrix. We call this matrix the Partition-Frequency Enumer
ation (PFE) matrix. This matrix unifies a large number of results connecti
ng number-theoretic functions to partition-type functions. The calculus is
extended to arbitrary generating functions\, and functions with Weierstra
ss products. As a by-product\, we recover (and extend) some well-known rec
urrence relations for many number-theoretic functions\, including the sum
of divisors function\, Ramanujan's $\\tau$ function\, sums of squares and
triangular numbers\, and for $\\zeta(2n)$\, where $n$ is a positive intege
r. These include classical results due to Euler\, Ramanujan\, and others.
As one application\, we embed Ramanujan's famous congruences $p(5n+4)\\eq
uiv 0$ (mod $5)$ and $\\tau(5n+5)\\equiv 0$ (mod $5)$.\n\nThis is joint wo
rk with Hartosh Singh Bal.\ninto an infinite family of such congruences.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liuquan Wang (Wuhan University)
DTSTART:20210304T103000Z
DTEND:20210304T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/19
DESCRIPTION:Title: Parity of coefficients of mock theta functions\nby Liuquan W
ang (Wuhan University) as part of Special Functions and Number Theory semi
nar\n\n\nAbstract\nWe study the parity of coefficients of classical mock t
heta functions. Suppose $g$ is a formal power series with integer coeffici
ents\, and let $c(g\;n)$ be the coefficient of $q^n$ in its series expansi
on. We say that $g$ is of parity type $(a\,1-a)$ if $c(g\;n)$ takes even v
alues with probability $a$ for $n\\geq 0$. We show that among the 44 class
ical mock theta functions\, 21 of them are of parity type $(1\,0)$. We fur
ther conjecture that 19 mock theta functions are of parity type $(\\frac{1
}{2}\,\\frac{1}{2})$ and 4 functions are of parity type $(\\frac{3}{4}\,\\
frac{1}{4})$. We also give characterizations of $n$ such that $c(g\;n)$ is
odd for the mock theta functions of parity type $(1\,0)$.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Krattenthaler (University of Vienna\, Austria)
DTSTART:20210318T103000Z
DTEND:20210318T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/20
DESCRIPTION:Title: Determinant identities for moments of orthogonal polynomials
\nby Christian Krattenthaler (University of Vienna\, Austria) as part of S
pecial Functions and Number Theory seminar\n\n\nAbstract\nWe present a for
mula that expresses the Hankel determinants of a linear combination of len
gth d+1 of moments of orthogonal polynomials in terms of a d x d determina
nt of the orthogonal polynomials. As a literature search\nrevealed\, this
formula exists somehow hidden in the folklore of the\ntheory of orthogonal
polynomials as it is related to "Christoffel's\ntheorem". In any case\, i
t deserves to be better known and be presented\ncorrectly and with full pr
oof. (During the talk I will explain the\nmeaning of these somewhat crypti
c formulations.) Subsequently\, I\nwill show an application of the formula
. I will close the talk by\npresenting a generalisation that is inspired b
y Uvarov's formula\nfor the orthogonal polynomials of rationally related d
ensities.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shishuo Fu (Chongqing University)
DTSTART:20210401T103000Z
DTEND:20210401T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/21
DESCRIPTION:Title: Bijective recurrences for Schroeder triangles and Comtet statist
ics\nby Shishuo Fu (Chongqing University) as part of Special Functions
and Number Theory seminar\n\n\nAbstract\nIn this talk\, we bijectively es
tablish recurrence relations for two triangular arrays\, relying on their
interpretations in terms of Schroeder paths (resp. little Schroeder paths)
with given length and number of hills. The row sums of these two triangle
s produce the large (resp. little) Schroeder numbers. On the other hand\,
it is well-known that the large Schroeder numbers also enumerate separable
permutations. This propelled us to reveal the connection with a lesser-kn
own permutation statistic\, called initial ascending run (iar)\, whose dis
tribution on separable permutations is shown to be given by the first tria
ngle as well. A by-product of this result is that "iar" is equidistributed
over separable permutations with "comp"\, the number of components of a p
ermutation. We call such statistics Comtet and we briefly mention further
work concerning Comtet statistics on various classes of pattern avoiding p
ermutations. The talk is based on joint work with Zhicong Lin and Yaling W
ang.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nayandeep Deka Baruah (Tezpur University)
DTSTART:20210415T103000Z
DTEND:20210415T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/22
DESCRIPTION:Title: Matching coefficients in the series expansions of certain $q$-pr
oducts and their inverses\nby Nayandeep Deka Baruah (Tezpur University
) as part of Special Functions and Number Theory seminar\n\n\nAbstract\nWe
show that the series expansions of certain $q$-products have \\textit{ma
tching coefficients} with their inverses. Several of the results are assoc
iated to Ramanujan's continued fractions. For example\, let $R(q)$ denote
the Rogers-Ramanujan continued fraction having the well-known $q$-product
repesentation $R(q)=\\left(q\,q^4\;q^5\\right)_{\\infty}/\\left(q^2\,q^3\;
q^5\\right)_{\\infty}$. If\n\\begin{align*}\n\\sum_{n=0}^{\\infty}\\alpha(
n)q^n=\\dfrac{1}{R^5\\left(q\\right)}=\\left(\\sum_{n=0}^{\\infty}\\alpha^
{\\prime}(n)q^n\\right)^{-1}\,\\\\\n\\sum_{n=0}^{\\infty}\\beta(n)q^n=\\df
rac{R(q)}{R\\left(q^{16}\\right)}=\\left(\\sum_{n=0}^{\\infty}\\beta^{\\pr
ime}(n)q^n\\right)^{-1}\,\n\\end{align*}\nthen\n\\begin{align*}\n\\alpha(5
n+r)&=-\\alpha^{\\prime}(5n+r-2)\, \\quad r\\in\\{3\,4\\}\\\\\n\\text{and}
&\\\\\n\\beta(10n+r)&=-\\beta^{\\prime}(10n+r-6)\, \\quad r\\in\\{7\,9\\}.
\n\\end{align*}\nThis is a joint work with Hirakjyoti Das.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ankush Goswami (IIT Gandhinagar)
DTSTART:20210527T103000Z
DTEND:20210527T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/23
DESCRIPTION:Title: Partial theta series with periodic coefficients and quantum modu
lar forms\nby Ankush Goswami (IIT Gandhinagar) as part of Special Func
tions and Number Theory seminar\n\n\nAbstract\nTheta series first appeare
d in Euler’s work on partitions\, but was systematically studied later b
y Jacobi. In his Lost Notebook\, Ramanujan wrote down many identities (wi
thout proof) involving the so-called partial theta series. Unlike the thet
a series which are modular forms\, the theory of partial theta series is n
ot well understood. In this talk\, I will consider a family of partial the
ta series and show their “quantum modular” behaviour. This is based on
my recent joint work with Robert Osburn (UCD).\n\nThe talk should be acce
ssible to graduate and advanced undergraduate students.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritabrata Munshi (ISI\, Kolkata)
DTSTART:20210610T103000Z
DTEND:20210610T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/24
DESCRIPTION:Title: 100 years of sub-convexity\nby Ritabrata Munshi (ISI\, Kolka
ta) as part of Special Functions and Number Theory seminar\n\n\nAbstract\n
I will present a historical survey of the sub-convexity problem.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debika Banerjee (IIIT\, Delhi)
DTSTART:20210624T103000Z
DTEND:20210624T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/25
DESCRIPTION:Title: Bessel functions and their application to classical number theor
y\nby Debika Banerjee (IIIT\, Delhi) as part of Special Functions and
Number Theory seminar\n\n\nAbstract\nFinding solutions of differential equ
ations has been a problem in pure mathematics since the invention of calcu
lus by Newton and Leibniz in the 17th century. Bessel functions are solut
ions of a particular differential equation\, called Bessel’s equation. I
n classical analytic number theory\, there are several summation formulas
or trace formulas involving Bessel functions. Two prominent such are the K
uznetsov trace formula and the Voronoi summation formula. In this talk\, I
will present some Voronoi type summation formulas and its application to
Number theory.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bibekananda Maji (IIT\, Indore)
DTSTART:20210708T103000Z
DTEND:20210708T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/26
DESCRIPTION:Title: On Ramanujan's formula for $\\zeta(1/2)$ and $\\zeta(2m+1)$\
nby Bibekananda Maji (IIT\, Indore) as part of Special Functions and Numbe
r Theory seminar\n\n\nAbstract\nEuler's remarkable formula for $\\zeta(2m)
$ immediately tells us that even zeta values are transcendental. However\,
the algebraic nature of odd zeta values is yet to be determined. \nPage
320 and 332 of Ramanujan's Lost Notebook contains an intriguing identity f
or $\\zeta(2m+1)$ and $\\zeta(1/2)$\, respectively. Many mathematicians h
ave studied these identities over the years.\n\nIn this talk\, we shall di
scuss transformation formulas for a certain infinite series\, which will
enable us to derive Ramanujan's formula for $\\zeta(1/2)\,$ Wigert's formu
la for $\\zeta(1/k)$\, as well as Ramanujan's formula for $\\zeta(2m+1)$.
We also obtain a new identity for $\\zeta(-1/2)$ in the spirit of Ramanuja
n.\n\nThis is joint work with Anushree Gupta.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anup Biswanath Dixit (IMSc (Chennai))
DTSTART:20210722T103000Z
DTEND:20210722T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/27
DESCRIPTION:Title: On Euler-Kronecker constants and the class number problem\nb
y Anup Biswanath Dixit (IMSc (Chennai)) as part of Special Functions and N
umber Theory seminar\n\n\nAbstract\nAs a natural generalization of the Eul
er's constant \n$\\gamma$\, Y. Ihara introduced the Euler-Kronecker const
ants attached \nto any number field. In this talk\, we will discuss the c
onnection \nbetween these constants and certain arithmetic properties of
number \nfields.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter A Clarkson (University of Kent\, Canterbury\, UK)
DTSTART:20210805T103000Z
DTEND:20210805T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/28
DESCRIPTION:Title: Special polynomials associated with the Painlev\\'{e} equations<
/a>\nby Peter A Clarkson (University of Kent\, Canterbury\, UK) as part of
Special Functions and Number Theory seminar\n\n\nAbstract\nThe six Painle
v\\'{e} equations\, whose solutions are called the Painlev\\'{e} transcend
ents\, were derived by Painlev\\'{e} and his colleagues in the late 19th a
nd early 20th centuries in a classification of second order ordinary diffe
rential equations whose solutions have no movable critical points.\nIn the
18th and 19th centuries\, the classical special functions such as Bessel\
, Airy\, Legendre and hypergeometric functions\, were recognized and devel
oped in response to the problems of the day in electromagnetism\, acoustic
s\, hydrodynamics\, elasticity and many other areas.\nAround the middle of
the 20th century\, as science and engineering continued to expand in new
directions\, a new class of functions\, the Painlev\\'{e} functions\, star
ted to appear in applications. The list of problems now known to be descri
bed by the Painlev\\'{e} equations is large\, varied and expanding rapidly
. The list includes\, at one end\, the scattering of neutrons off heavy nu
clei\, and at the other\, the distribution of the zeros of the Riemann-zet
a function on the critical line $\\mbox{Re}(z) =\\tfrac12$. Amongst many o
thers\, there is random matrix theory\, the asymptotic theory of orthogona
l polynomials\, self-similar solutions of integrable equations\, combinato
rial problems such as the longest increasing subsequence problem\, tiling
problems\, multivariate statistics in the important asymptotic regime wher
e the number of variables and the number of samples are comparable and lar
ge\, and also random growth problems.\n\nThe Painlev\\'{e} equations posse
ss a plethora of interesting properties including a Hamiltonian structure
and associated isomonodromy problems\, which express the Painlev\\'{e} equ
ations as the compatibility condition of two linear systems. Solutions of
the Painlev\\'{e} equations have some interesting asymptotics which are us
eful in applications. They possess hierarchies of rational solutions and o
ne-parameter families of solutions expressible in terms of the classical s
pecial functions\, for special values of the parameters. Further the Painl
ev\\'{e} equations admit symmetries under affine Weyl groups which are rel
ated to the associated B\\"ack\\-lund transformations.\n\nIn this talk I s
hall discuss special polynomials associated with rational solutions of Pai
nlev\\'{e} equations. Although the general solutions of the six Painlev\\'
{e} equations are transcendental\, all except the first Painlev\\'{e} equa
tion possess rational solutions for certain values of the parameters. Thes
e solutions are expressed in terms of special polynomials\nThe roots of th
ese special polynomials are highly symmetric in the complex plane and spec
ulated to be of interest to number theorists. The polynomials arise in app
lications such as random matrix theory\, vortex dynamics\, in supersymmetr
ic quantum mechanics\, as coefficients of recurrence relations for semi-cl
assical orthogonal polynomials and are examples of exceptional orthogonal
polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter A Clarkson (University of Kent\, Canterbury\, UK)
DTSTART:20210805T103000Z
DTEND:20210805T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/29
DESCRIPTION:Title: Special polynomials associated with the Painlev\\'{e} equations<
/a>\nby Peter A Clarkson (University of Kent\, Canterbury\, UK) as part of
Special Functions and Number Theory seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajat Gupta (IIT\, Gandhinagar)
DTSTART:20210819T103000Z
DTEND:20210819T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/30
DESCRIPTION:Title: Koshliakov zeta functions and modular relations\nby Rajat Gu
pta (IIT\, Gandhinagar) as part of Special Functions and Number Theory sem
inar\n\n\nAbstract\nNikolai Sergeevich Koshliakov was an outstanding Russi
an mathematician who made phenomenal contributions to number theory and di
fferential equations. In the aftermath of World War II\, he was one among
the many scientists who were arrested on fabricated charges and incarcerat
ed. Under extreme hardships while still in prison\, Koshliakov (under a di
fferent name `N. S. Sergeev') wrote two manuscripts out of which one was l
ost. Fortunately the second one was published in 1949 although\, to the be
st of our knowledge\, no one studied it until the last year when Prof. Atu
l Dixit and I started examining it in detail. This manuscript contains a c
omplete theory of two interesting generalizations of the Riemann zeta func
tion having their genesis in heat conduction and is truly a masterpiece! I
n this talk\, we will discuss some of the contents of this manuscript and
then proceed to give some new results (modular relations) that we have obt
ained in this theory. This is joint work with Prof. Atul Dixit.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Howard Cohl (NIST)
DTSTART:20210902T130000Z
DTEND:20210902T140000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/31
DESCRIPTION:Title: The utility of integral representations for the Askey-Wilson pol
ynomials and their symmetric sub-families\nby Howard Cohl (NIST) as pa
rt of Special Functions and Number Theory seminar\n\n\nAbstract\nThe Askey
-Wilson polynomials are a class of orthogonal polynomials which are symmet
ric in four free parameters which lie at the very top of the q-Askey schem
e of basic hypergeometric orthogonal polynomials. These polynomials\, and
the polynomials in their subfamilies\, are usually defined in terms of the
ir finite series representations which are given in terms of terminating b
asic hypergeometric series. However\, they also have nonterminating\, q-in
tegral\, and integral representations. In this talk\, we will explore some
of what is known about the symmetry of these representations and how they
have been used to compute their important properties such as generating
functions. This study led to an extension of interesting contour integral
representations of sums of nonterminating basic hypergeometric functions i
nitially studied by Bailey\, Slater\, Askey\, Roy\, Gasper and Rahman. We
will also discuss how these contour integrals are deeply connected to the
properties of the symmetric basic hypergeometric orthogonal polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neelam Saikia (University of Virginia)
DTSTART:20210916T103000Z
DTEND:20210916T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/32
DESCRIPTION:Title: Frobenius trace distributions for Gaussian hypergeometric functi
ons\nby Neelam Saikia (University of Virginia) as part of Special Func
tions and Number Theory seminar\n\n\nAbstract\nIn the 1980's\, Greene defi
ned hypergeometric functions over finite fields using Jacobi sums. These f
unctions possess many properties that are analogous to those of the classi
cal hypergeometric series studied by Gauss\, Kummer and others. These func
tions have played important roles in the study of supercongruences\, the E
ichler-Selberg trace formula\, and zeta-functions of arithmetic varieties.
We study the distribution (over large finite fields) of the values of cer
tain families of these functions. For the $_2F_1$ functions\, the limiting
distribution is semicircular\, whereas the distribution for the $_3F_2$ f
unctions is the more exotic \\it{Batman distribution.}\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaban Meher (NISER\, Bhubaneswar)
DTSTART:20210930T103000Z
DTEND:20210930T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/33
DESCRIPTION:Title: Modular forms and certain congruences\nby Jaban Meher (NISER
\, Bhubaneswar) as part of Special Functions and Number Theory seminar\n\n
\nAbstract\nIn this talk we shall discuss about modular forms and certain
types of congruences among the Fourier coefficients of modular forms. We s
hall also discuss about the non-existence of Ramanujan-type congruences fo
r certain modular forms.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meesue Yoo (Chungbuk National University\, Korea.)
DTSTART:20211014T103000Z
DTEND:20211014T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/34
DESCRIPTION:Title: Elliptic rook and file numbers\nby Meesue Yoo (Chungbuk Nati
onal University\, Korea.) as part of Special Functions and Number Theory s
eminar\n\n\nAbstract\nIn this talk\, we construct elliptic analogues of th
e rook numbers and file numbers by attaching elliptic weights to the cells
in a board. We show that our elliptic rook and file numbers satisfy ellip
tic extensions of corresponding factorization theorems which in the classi
cal case was established by Goldman\, Joichi and White and by Garsia and R
emmel in the file number case. This factorization theorem can be used to d
efine elliptic analogues of various kinds of Stirling numbers of the first
and second kind\, and Abel numbers. \n\n\nWe also give analogous results
for matchings of graphs\, elliptically extending the result of Haglund and
Remmel.\n\n\nThis is joint work with Michael Schlosser.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshaa Vatwani (IIT\, Gandhinagar)
DTSTART:20211028T103000Z
DTEND:20211028T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/35
DESCRIPTION:Title: Limitations to equidistribution in arithmetic progressions\n
by Akshaa Vatwani (IIT\, Gandhinagar) as part of Special Functions and Num
ber Theory seminar\n\n\nAbstract\nIt is well known that the prime numbers
are equidistributed in arithmetic progressions. Such a phenomenon is also
observed more generally for a class of arithmetic functions. A key result
in this context is the Bombieri-Vinogradov theorem which establishes that
the primes are equidistributed in arithmetic progressions ``on average" fo
r moduli $q$ in the range $q \\le x^{1/2 -\\epsilon }$ for any $\\epsilon>
0$. In 1989\, building on an idea of Maier\, Friedlander and Granville sho
wed that such equidistribution results fail if the range of the moduli $q$
is extended to $q \\le x/ (\\log x)^B $ for any $B>1$. We discuss variant
s of this result and give some applications. This is joint work with Aditi
Savalia.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kamalakshya Mahatab (Chennai Mathematical Institute (CMI))
DTSTART:20211111T103000Z
DTEND:20211111T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/36
DESCRIPTION:Title: Large oscillations of the argument of the Riemann zeta function<
/a>\nby Kamalakshya Mahatab (Chennai Mathematical Institute (CMI)) as part
of Special Functions and Number Theory seminar\n\n\nAbstract\nWe will obt
ain large values of the argument of the Riemann zeta function using the re
sonance method. We will also apply the method to the iterated arguments. T
his is a joint work with A. Chirre\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Koustav Banerjee (RISC\, JKU\, Linz)
DTSTART:20211125T103000Z
DTEND:20211125T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/37
DESCRIPTION:Title: Inequalities for the modified Bessel function of first kind and
its consequences\nby Koustav Banerjee (RISC\, JKU\, Linz) as part of S
pecial Functions and Number Theory seminar\n\n\nAbstract\nStudy on asympto
tics of modified Bessel functions dates back to 18th century.\nIn this tal
k\, we will describe how from the study of asymptotics of modified Bessel
function of\nfirst kind of non-negative order\, one can comes up with a ho
st of inequalities that finally leads\nto answer combinatorial properties\
, for example log-concavity\, higher order Tur\\acute{a}n inequality\nof c
ertain arithmetic sequences arising from Fourier coefficients of modular f
orms. In addition\nto that\, we will discuss briefly on a result of Bringm
ann et al. and analyze with the work\naddressed above.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Sokal (University College\, London and NYU)
DTSTART:20030127T103000Z
DTEND:20030127T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/38
DESCRIPTION:Title: Coefficientwise Hankel-total positivity\nby Alan Sokal (Univ
ersity College\, London and NYU) as part of Special Functions and Number T
heory seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Sokal (University College\, London and NYU)
DTSTART:20220127T103000Z
DTEND:20220127T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/39
DESCRIPTION:Title: Coefficientwise Hankel-total positivity\nby Alan Sokal (Univ
ersity College\, London and NYU) as part of Special Functions and Number T
heory seminar\n\n\nAbstract\nA matrix $M$ of real numbers is called {\\em
totally positive}\\/\n if every minor of $M$ is nonnegative. Gantmakher
and Krein showed\n in 1937 that a Hankel matrix $H = (a_{i+j})_{i\,j \\
ge 0}$\n of real numbers is totally positive if and only if the underlyi
ng\n sequence $(a_n)_{n \\ge 0}$ is a Stieltjes moment sequence.\n Mor
eover\, this holds if and only if the ordinary generating function\n $\\
sum_{n=0}^\\infty a_n t^n$ can be expanded as a Stieltjes-type\n continu
ed fraction with nonnegative coefficients:\n$$\n \\sum_{n=0}^{\\infty} a
_n t^n\n \\\;=\\\;\n \\cfrac{\\alpha_0}{1 - \\cfrac{\\alpha_1 t}{1 - \
\cfrac{\\alpha_2 t}{1 - \\cfrac{\\alpha_3 t}{1- \\cdots}}}}\n$$\n (in t
he sense of formal power series) with all $\\alpha_i \\ge 0$.\n So total
ly positive Hankel matrices are closely connected with\n the Stieltjes m
oment problem and with continued fractions.\n\n Here I will introduce a
generalization: a matrix $M$ of polynomials\n (in some set of indetermi
nates) will be called\n {\\em coefficientwise totally positive}\\/ if ev
ery minor of $M$\n is a polynomial with nonnegative coefficients. And
a sequence\n $(a_n)_{n \\ge 0}$ of polynomials will be called\n {\\em
coefficientwise Hankel-totally positive}\\/ if the Hankel matrix\n $H =
(a_{i+j})_{i\,j \\ge 0}$ associated to $(a_n)$ is coefficientwise\n tot
ally positive. It turns out that many sequences of polynomials\n arisin
g naturally in enumerative combinatorics are (empirically)\n coefficient
wise Hankel-totally positive. In some cases this can\n be proven using
continued fractions\, by either combinatorial or\n algebraic methods\;
I will sketch how this is done. In many other\n cases it remains an ope
n problem.\n\n One of the more recent advances in this research is perha
ps of\n independent interest to special-functions workers:\n we have f
ound branched continued fractions for ratios of contiguous\n hypergeomet
ric series ${}_r \\! F_s$ for arbitrary $r$ and $s$\,\n which generalize
Gauss' continued fraction for ratios of contiguous\n ${}_2 \\! F_1$. F
or the cases $s=0$ we can use these to prove\n coefficientwise Hankel-to
tal positivity.\n\n Reference: Mathias P\\'etr\\'eolle\, Alan D.~Sokal a
nd Bao-Xuan Zhu\,\n arXiv:1807.03271\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Surbhi Rai (IIT\, Delhi)
DTSTART:20220210T103000Z
DTEND:20220210T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/40
DESCRIPTION:Title: Expansion Formulas for Multiple Basic Hypergeometric Series Ove
r Root Systems\nby Surbhi Rai (IIT\, Delhi) as part of Special Functio
ns and Number Theory seminar\n\n\nAbstract\nIn a series of works\, Zhi-Guo
Liu extended some of the central summation and transformation formulas of
basic hypergeometric series. In particular\, Liu extended Rogers' non-ter
minating very-well-poised $_{6}\\phi_{5}$ summation formula\, Watson's t
ransformation\nformula\, and gave an alternate approach to the orthogonali
ty of the Askey-Wilson polynomials. These results are helpful in number-th
eoretic contexts too. All this work relies on three expansion formulas of
Liu.\n\nThis talk will present several infinite families of extensions of
Liu's fundamental formulas to multiple basic hypergeometric series over ro
ot systems. We will also discuss results that extend Wang and Ma's general
izations of Liu's work which they obtained using $q$-Lagrange inversion. S
ubsequently\, we will look at an application based on the expansions of in
finite products. These extensions have been obtained using the $A_n$ and $
C_n$ Bailey transformation and other summation theorems due to Gustafson\,
Milne\, Milne and Lilly\, and others\, from $A_n$\, $C_n$ and $D_n$ basic
hypergeometric series theory. We will observe how this approach brings Li
u's expansion formulas within the Bailey transform methodology.\n\nThis ta
lk is based on joint work with Gaurav Bhatnagar. (https://arxiv.org/abs/21
09.02827)\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krishnan Rajkumar (JNU)
DTSTART:20220224T103000Z
DTEND:20220224T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/41
DESCRIPTION:Title: The Binet function and telescoping continued fractions\nby K
rishnan Rajkumar (JNU) as part of Special Functions and Number Theory semi
nar\n\n\nAbstract\nThe Binet function $J(z)$ defined by the equation $\\Ga
mma(z) = \\sqrt{2 \\pi} z^{z-\\frac{1}{2}}e^{-z} e^{J(z)}$ is a well-stud
ied function. The Stirling approximation comes from the property $J(z) \\r
ightarrow 0$ as $z\\rightarrow \\infty$\, $|arg z|<\\pi$. In fact\, an asy
mptotic expansion $J(z) \\sim z^{-1} \\sum_{k=0}^{\\infty} c_k z^{-2k}$ ho
lds in this region\, with closed form expressions for $c_k$ and explicit i
ntegrals for the error term for any finite truncation of this asymptotic s
eries. \n\nIn this talk\, we will discuss two different classical directio
ns of research. The first is exemplified by the work of Robbins (1955) and
Cesaro (1922)\, and carried forward by several authors\, the latest being
Popov (2018)\, where elementary means are used to find rational lower and
upper bounds for $J(n)$ which hold for all positive integers $n$. All of
these establish inequalities of the form $J(n)-J(n+1) > F(n)-F(n+1)$ for a
n appropriate rational function $F$ to derive the corresponding lower boun
ds by telescoping.\n\nThe second direction is to use moment theory to deri
ve continued fractions of specified forms for $J(x)$. For instance\, a mod
ified S-fraction of the form $\\frac{a_1}{x \\ +} \\frac{a_2}{x \\ +}\\fra
c{a_3}{x \\ +} \\cdots$ can be formally derived from the above asymptotic
expansion using a method called the qd-algorithm. The resulting continued
fraction can then be shown to converge to $J(x)$ by the asymptotic propert
ies of $c_k$ and powerful results from moment theory. There are no known c
losed-form expressions for the $a_k$.\n\nWe will then outline what we call
the method of telescoping continued fractions to extend the elementary me
thods of the first approach to derive the modified S-fraction for $J(x)$ o
btained in the second by a new algorithm. We will describe several results
that we can prove and some conjectures that together enhance our understa
nding of the numbers $a_k$ as well as provide upper and lower bounds for $
J(x)$ that improve all known results.\n\nThis is joint work with Gaurav Bh
atnagar.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenguang Zhai (China Institute of Mining and Technology\, Beijing\
, PRC)
DTSTART:20220310T103000Z
DTEND:20220310T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/42
DESCRIPTION:Title: The MC-algorithm and continued fraction formulas involving ratio
s of Gamma functions\nby Wenguang Zhai (China Institute of Mining and
Technology\, Beijing\, PRC) as part of Special Functions and Number Theory
seminar\n\n\nAbstract\nRamanujan discovered many continued fraction expan
sions about ratios of the Gamma functions. However\, Ramanujan left us no
clues about how he discovered these elegant formulas. In this talk\, we wi
ll explain the so-called MC-algorithm. By this algorithm\, we can not only
rediscover many of Ramanujan's continued fraction expansions\, but also f
ind some new formulas.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumyarup Banerjee (IIT\, Gandhinagar)
DTSTART:20220324T103000Z
DTEND:20220324T113000Z
DTSTAMP:20260404T111333Z
UID:SF-and-nt/43
DESCRIPTION:Title: Finiteness theorems with almost prime inputs\nby Soumyarup B
anerjee (IIT\, Gandhinagar) as part of Special Functions and Number Theory
seminar\n\n\nAbstract\nThe Conway–Schneeberger Fifteen theorem states t
hat a given positive definite integral quadratic form is universal (i.e.\,
represents every positive integer with integer inputs) if and only if it
represents the integers up to 15. This theorem is sometimes known as “Fi
niteness theorem" as it reduces an infinite check to a finite one. In this
talk\, I would like to present my recent work along with Ben Kane where I
have investigated quadratic forms which are universal when restricted to
almost prime inputs and have established finiteness theorems akin to the C
onway–Schneeberger Fifteen theorem.\n
LOCATION:https://stable.researchseminars.org/talk/SF-and-nt/43/
END:VEVENT
END:VCALENDAR