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BEGIN:VEVENT
SUMMARY:Arvind Ayyer (IISc\, Bangalore)
DTSTART;VALUE=DATE-TIME:20200618T103000Z
DTEND;VALUE=DATE-TIME:20200618T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
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://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;VALUE=DATE-TIME:20200702T092500Z
DTEND;VALUE=DATE-TIME:20200702T103000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/2
DESCRIPTION:Title: On the Kanade-Russell identities\nby Hjalmar Rosengren (Chalmers Uni
versity of Technology and the University of Gothenburg) as part of Special
Functions and Number Theory seminar\n\n\nAbstract\nKanade and Russell con
jectured several Rogers-Ramanujan-type identities for triple series. Some
of these conjectures are related to characters of affine Lie algebras\, an
d they can all be interpreted combinatorially in terms of partitions. Many
of these conjectures were settled by Bringmann\, Jennings-Shaffer and Mah
lburg. 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 quadratic tr
ansformations for q-orthogonal polynomials.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ali Uncu (RICAM\, Austrian Academy of Sciences)
DTSTART;VALUE=DATE-TIME:20200716T093000Z
DTEND;VALUE=DATE-TIME:20200716T103000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/3
DESCRIPTION:Title: The Mathematica package qFunctions for q-series and partition theory app
lications\nby Ali Uncu (RICAM\, Austrian Academy of Sciences) as part
of Special Functions and Number Theory seminar\n\n\nAbstract\nIn this talk
\, I will demonstrate the new Mathematica package qFunctions while providi
ng relevant mathematical context. This implementation has symbolic tools t
o automate some tedious and error-prone calculations and it also includes
some other functionality for experimentation. We plan to highlight the fou
r main tool-sets included in the qFunctions package:\n\n(1) The q-differen
ce equation (or recurrence) guesser and some formal manipulation tools\,\n
(2) the treatment of the method of weighted words and automatically findin
g and uncoupling recurrences\,\n\n(3) a method on the cylindrical partitio
ns to establish sum-product identities\,\n(4) fitting polynomials with sug
gested well-known objects to guess closed formulas.\n\nThis talk is based
on joint work with Jakob Ablinger (RISC).\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alok Shukla (Ahmedabad University)
DTSTART;VALUE=DATE-TIME:20200730T102500Z
DTEND;VALUE=DATE-TIME:20200730T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/4
DESCRIPTION:Title: Tiling proofs of Jacobi triple product and Rogers-Ramanujan identities\nby Alok Shukla (Ahmedabad University) as part of Special Functions and
Number Theory seminar\n\n\nAbstract\nThe Jacobi triple product identity a
nd Rogers-Ramanujan identities are among the most famous q-series identiti
es. We will present elementary combinatorial "tiling proofs" of these resu
lts. The talk should be accessible to a general mathematical audience.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fatma Cicek (IIT\, Gandhinagar)
DTSTART;VALUE=DATE-TIME:20200813T103000Z
DTEND;VALUE=DATE-TIME:20200813T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/5
DESCRIPTION:Title: On the logarithm of the Riemann zeta-function near the nontrivial zeros<
/a>\nby Fatma Cicek (IIT\, Gandhinagar) as part of Special Functions and N
umber Theory seminar\n\n\nAbstract\nSelberg's central limit theorem is one
of the most significant probabilistic results in analytic number theory.
Roughly\, it states that the logarithm of the Riemann zeta-function on and
near the critical line has an approximate two-dimensional Gaussian distri
bution.\n\nIn this talk\, we will talk about our recent result which state
s that the distribution of the logarithm of the Riemann zeta-function near
the sequence of the nontrivial zeros has a similar central limit theorem.
Our results are conditional on the Riemann Hypothesis and/or suitable zer
o-spacing hypotheses. They also have suitable generalizations to Dirichlet
$L$-functions.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amritanshu Prasad (IMSc\, Chennai)
DTSTART;VALUE=DATE-TIME:20200827T103000Z
DTEND;VALUE=DATE-TIME:20200827T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
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\nAbs
tract\nA polynomial in a sequence of variables can be regarded as a class
\nfunction on every symmetric group when the $i$th variable is interpreted
as \nthe number of $i$-cycles. Many nice families of representations of s
ymmetric \ngroups have characters represented by such polynomials.\n\nWe i
ntroduce two families linear functionals of this space of polynomials -- m
oments and signed moments. For each $n$\, the moment of a polynomial 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 \n
product by the sign character. These linear functionals are easy to \ncomp
ute in terms of binomial bases of the space of polynomials.\n\nWe use them
to explore some questions in the representation theory of \nsymmetric gro
ups and general linear groups. These explorations lead to \ninteresting ex
pressions involving multipartite partition functions and \nsome peculiar v
ariants.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (IISc.\, Bangalore)
DTSTART;VALUE=DATE-TIME:20200910T103000Z
DTEND;VALUE=DATE-TIME:20200910T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/7
DESCRIPTION:Title: An introduction to total positivity\nby Apoorva Khare (IISc.\, Banga
lore) as part of Special Functions and Number Theory seminar\n\n\nAbstract
\nI will give a gentle introduction to total positivity and the theory of
Polya frequency (PF) functions. This includes their spectral properties\,
basic examples including via convolution\, and a few proofs to show how th
e main ingredients fit together. Many classical results (and one Hypothesi
s) from before 1955 feature in this journey. I will end by describing how
PF functions connect to the Laguerre-Polya class and hence Polya-Schur mul
tipliers\, and mention 21st century incarnations of the latter.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (IISc.\, Bangalore)
DTSTART;VALUE=DATE-TIME:20200924T103000Z
DTEND;VALUE=DATE-TIME:20200924T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/8
DESCRIPTION:Title: Totally positive matrices\, Polya frequency sequences\, and Schur polyno
mials\nby Apoorva Khare (IISc.\, Bangalore) as part of Special Functio
ns and Number Theory seminar\n\n\nAbstract\nI will discuss totally positiv
e/non-negative matrices and kernels\, including Polya frequency (PF) funct
ions and sequences. This includes examples\, history\, and basic results o
n total positivity\, variation diminution\, sign non-reversal\, and genera
ting functions of PF sequences (with some proofs). I will end with applica
tions of total positivity to old and new phenomena involving Schur polynom
ials.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debashis Ghoshal (School of Physical Sciences\, JNU)
DTSTART;VALUE=DATE-TIME:20201008T103000Z
DTEND;VALUE=DATE-TIME:20201008T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/9
DESCRIPTION:Title: Two-dimensional gauge theories\, intersection numbers and special functi
ons\nby Debashis Ghoshal (School of Physical Sciences\, JNU) as part o
f Special Functions and Number Theory seminar\n\n\nAbstract\nThe partition
function of two dimensional Yang-Mills theory contains a wealth of inform
ation about the moduli space of connections on surfaces. We study this pro
blem on a special class of surfaces of infinite genus\, which are construc
ted recursively. While the results are suggestive of an underlying geometr
ical structure\, we use it as a prop to efficiently compute results for fi
nite genus surfaces. Riemann zeta function\, confluent hypergeometric func
tion and its truncations show up in explicit computations for the gauge gr
oup SU(2). Much of the corresponding results are open for other groups.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Schlosser (Universitat Wien)
DTSTART;VALUE=DATE-TIME:20201022T103000Z
DTEND;VALUE=DATE-TIME:20201022T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/10
DESCRIPTION:Title: Basic hypergeometric proofs of two quadruple equidistributions of Euler
--Stirling statistics on ascent sequences\nby Michael Schlosser (Unive
rsitat Wien) as part of Special Functions and Number Theory seminar\n\n\nA
bstract\nIn my talk\, I will present new applications of basic\nhypergeome
tric series to specific problems in enumerative\ncombinatorics. The combin
atorial problems we are interested in\nconcern multiply refined equidistri
butions on ascent sequences.\n(I will gently explain these notions in my t
alk!)\nUsing bijections we are able to suitably decompose some\nquadruple
distributions we are interested in and obtain\nfunctional equations and ul
timately generating functions\nfrom them\, in the form of explicit basic h
ypergeometric series\,\nThe problem of proving equidistributions then redu
ces to\napplying suitable transformations of basic hypergeometric series.\
nThe situation in our case however is tricky (caused by the\nfact how the
power series variable $r$ appears in the base\n$q=1-r$ of the respective b
asic hypergeometric series\; so being\ninterested in the generating functi
on in $r$ as a Maclaurin\nseries\, we are thus interested in the analytic
expansion of\nthe nonterminating basic hypergeometric series in base $q$\n
around the point $q=1$)\, as none of the known transformations\nappear to
directly work to settle our problems\; we require\nthe derivation of new i
dentities.\nSpecifically\, we use the classical Sears transformation and\n
apply some analytic tools to establish a new non-terminating\n${}_4\\phi_3
$ transformation formula of base $q$\, valid as\nan identity in a neighbor
hood around $q=1$. We use special\ncases of this formula to deduce two dif
ferent quadruple\nequidistribution results involving Euler--Stirling stati
stics\non ascent sequences. One of them concerns a symmetric\nequidistrib
ution\, the other confirms a bi-symmetric\nequidistribution that was recen
tly conjectured in a paper\n(published in JCTA) by Shishuo Fu\, Emma Yu Ji
n\, Zhicong Lin\,\nSherry H.F. Yan\, and Robin D.B. Zhou. This is joint w
ork\nwith Emma Yu Jin. For full results (and further ones)\, see\nhttps://
arxiv.org/abs/2010.01435\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sanoli Gun (IMSc\, Chennai)
DTSTART;VALUE=DATE-TIME:20201112T103000Z
DTEND;VALUE=DATE-TIME:20201112T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/11
DESCRIPTION:Title: Large values of $L$-functions\nby Sanoli Gun (IMSc\, Chennai) as pa
rt of Special Functions and Number Theory seminar\n\n\nAbstract\nIn this l
ecture\, we will give an overview of a method of Soundararajan and show th
at how this method can be used to produce large values of $L$-functions in
different set-ups.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manjil P. Saikia (Cardiff University)
DTSTART;VALUE=DATE-TIME:20201126T103000Z
DTEND;VALUE=DATE-TIME:20201126T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/12
DESCRIPTION:Title: Refined enumeration of symmetry classes of Alternating Sign Matrices\nby Manjil P. Saikia (Cardiff University) as part of Special Functions a
nd Number Theory seminar\n\n\nAbstract\nThe sequence $1\,1\,2\,7\,42\,429\
, \\ldots$ counts several combinatorial objects\, some of which I will des
cribe in this talk. The major focus would be one of these objects\, altern
ating sign matrices (ASMs). ASMs are square matrices with entries in the s
et {0\,1\,-1}\, where non-zero entries alternate in sign along rows and co
lumns\, with all row and column sums being 1. I will discuss some question
s that are central to the theme of ASMs\, mainly dealing with their enumer
ation. In particular we shall prove some conjectures of Fischer\, Robbins\
, Duchon and Stroganov. This talk is based on joint work with Ilse Fischer
and some ongoing work.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sneha Chaubey (IIIT\, Delhi)
DTSTART;VALUE=DATE-TIME:20201210T103000Z
DTEND;VALUE=DATE-TIME:20201210T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
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 Number Theor
y seminar\n\n\nAbstract\nWe will discuss some subsets of two-dimensional i
nteger lattice which arise as visible sets under some suitable notion of v
isibility. We will discuss some set-theoretic (Delone\, Meyer\, Quasicryst
als etc.)\, geometrical (density and gaps) and dynamical (auto-correlation
and diffraction pattern) properties of these subsets\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rahul Kumar (IIT\, Gandhinagar)
DTSTART;VALUE=DATE-TIME:20201217T103000Z
DTEND;VALUE=DATE-TIME:20201217T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/14
DESCRIPTION:Title: A generalized modified Bessel function and explicit transformations of
certain Lambert series\nby Rahul Kumar (IIT\, Gandhinagar) as part of
Special Functions and Number Theory seminar\n\n\nAbstract\nAn exact transf
ormation\, which we call a master identity\, is obtained for the series $\
\sum_{n=1}^{\\infty}\\sigma_{a}(n)e^{-ny}$ for $a\\in\\mathbb{C}$ and Re$(
y)>0$. As corollaries when $a$ is an odd integer\, we derive the well-know
n transformations of the Eisenstein series on $\\textup{SL}_{2}\\left(\\ma
thbb{Z}\\right)$\, that of the Dedekind eta function as well as Ramanujan'
s famous formula for $\\zeta(2m+1)$. Corresponding new transformations whe
n $a$ is a non-zero even integer are also obtained as special cases of the
master identity. These include a novel companion to Ramanujan's formula f
or $\\zeta(2m+1)$. Although not modular\, it is surprising that such expli
cit 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 formul
a and an integral evaluation of N. S. Koshliakov involving a generalizatio
n of the modified Bessel function $K_{\\nu}(z)$. Koshliakov's integral eva
luation is proved for the first time. It is then generalized using a well-
known kernel of Watson to obtain an interesting two-variable generalizatio
n of the modified Bessel function. This generalization allows us to obtain
a new transformation involving the sums-of-squares function $r_k(n)$. Thi
s is joint work with Atul Dixit and Aashita Kesarwani.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wadim Zudilin (Radboud University)
DTSTART;VALUE=DATE-TIME:20210107T103000Z
DTEND;VALUE=DATE-TIME:20210107T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/15
DESCRIPTION:Title: Ramanujan special talk: 10 years of q-rious positivity. More needed!\nby Wadim Zudilin (Radboud University) as part of Special Functions and
Number Theory seminar\n\n\nAbstract\nThe $q$-binomial coefficients \\[ \\p
rod_{i=1}^m(1-q^{n-m+i})/(1-q^i)\,\\] for integers $0\\le m\\le n$\, are k
nown to be polynomials with non-negative integer coefficients. This readil
y follows from the $q$-binomial theorem\, or the many combinatorial interp
retations of them. Ten years ago\, together with Ole Warnaar we observed t
hat this non-negativity (aka positivity) property generalises to products
of ratios of $q$-factorials that happen to be polynomials\; we prove this
observation for (very few) cases. During the last decade a resumed interes
t in study of generalised integer-valued factorial ratios\, in connection
with problems in analytic number theory and combinatorics\, has brought to
life new positive structures for their $q$-analogues. In my talk I will r
eport on this "$q$-rious positivity" phenomenon\, an ongoing project with
Warnaar.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josef Kustner (University of Vienna)
DTSTART;VALUE=DATE-TIME:20210121T103000Z
DTEND;VALUE=DATE-TIME:20210121T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/16
DESCRIPTION:Title: Elliptic and $q$-analogs of the Fibonomial numbers\nby Josef Kustne
r (University of Vienna) as part of Special Functions and Number Theory se
minar\n\n\nAbstract\nThe Fibonomial numbers are integer numbers obtained f
rom the binomial coefficients by replacing each term by its corresponding
Fibonacci number. In 2009\, Sagan and Savage introduced a simple combinato
rial model for the Fibonomial numbers. \n\nIn this talk\, I will present a
combinatorial description for a q-analog and an elliptic analog of the Fi
bonomial numbers which is achieved by introducing certain q- and elliptic
weights to the model of Sagan and Savage.\n\nThis is joint work with Nante
l Bergeron and Cesar Ceballos.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victor Moll (Tulane University)
DTSTART;VALUE=DATE-TIME:20210204T103000Z
DTEND;VALUE=DATE-TIME:20210204T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/17
DESCRIPTION:Title: Valuations of interesting sequences\nby Victor Moll (Tulane Univers
ity) as part of Special Functions and Number Theory seminar\n\n\nAbstract\
nGiven a sequence ${ a_{n} }$ of integers and a prime $p$\, the sequence o
f\nvaluation $nu_{p}(a_{n})$ presents interesting challenges. This talk w
ill discuss a\nvariety of examples in order to illustrate these challenges
and present our approach\nto this problem.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gaurav Bhatnagar (Ashoka University)
DTSTART;VALUE=DATE-TIME:20210218T103000Z
DTEND;VALUE=DATE-TIME:20210218T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/18
DESCRIPTION:Title: The Partition-Frequency Enumeration Matrix\nby Gaurav Bhatnagar (As
hoka University) as part of Special Functions and Number Theory seminar\n\
n\nAbstract\nWe develop a calculus that gives an elementary approach to en
umerate partition-like objects using an infinite upper-triangular number-t
heoretic matrix. We call this matrix the Partition-Frequency Enumeration (
PFE) matrix. This matrix unifies a large number of results connecting numb
er-theoretic functions to partition-type functions. The calculus is extend
ed to arbitrary generating functions\, and functions with Weierstrass prod
ucts. As a by-product\, we recover (and extend) some well-known recurrence
relations for many number-theoretic functions\, including the sum of divi
sors function\, Ramanujan's $\\tau$ function\, sums of squares and triangu
lar numbers\, and for $\\zeta(2n)$\, where $n$ is a positive integer. Thes
e include classical results due to Euler\, Ramanujan\, and others. As one
application\, we embed Ramanujan's famous congruences $p(5n+4)\\equiv 0$
(mod $5)$ and $\\tau(5n+5)\\equiv 0$ (mod $5)$.\n\nThis is joint work with
Hartosh Singh Bal.\ninto an infinite family of such congruences.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liuquan Wang (Wuhan University)
DTSTART;VALUE=DATE-TIME:20210304T103000Z
DTEND;VALUE=DATE-TIME:20210304T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/19
DESCRIPTION:Title: Parity of coefficients of mock theta functions\nby Liuquan Wang (Wu
han University) as part of Special Functions and Number Theory seminar\n\n
\nAbstract\nWe study the parity of coefficients of classical mock theta fu
nctions. Suppose $g$ is a formal power series with integer coefficients\,
and let $c(g\;n)$ be the coefficient of $q^n$ in its series expansion. We
say that $g$ is of parity type $(a\,1-a)$ if $c(g\;n)$ takes even values w
ith probability $a$ for $n\\geq 0$. We show that among the 44 classical mo
ck theta functions\, 21 of them are of parity type $(1\,0)$. We further co
njecture 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 fo
r the mock theta functions of parity type $(1\,0)$.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christian Krattenthaler (University of Vienna\, Austria)
DTSTART;VALUE=DATE-TIME:20210318T103000Z
DTEND;VALUE=DATE-TIME:20210318T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/20
DESCRIPTION:Title: Determinant identities for moments of orthogonal polynomials\nby Ch
ristian Krattenthaler (University of Vienna\, Austria) as part of Special
Functions and Number Theory seminar\n\n\nAbstract\nWe present a formula th
at expresses the Hankel determinants of a linear combination of length d+1
of moments of orthogonal polynomials in terms of a d x d determinant of t
he orthogonal polynomials. As a literature search\nrevealed\, this formula
exists somehow hidden in the folklore of the\ntheory of orthogonal polyno
mials as it is related to "Christoffel's\ntheorem". In any case\, it deser
ves to be better known and be presented\ncorrectly and with full proof. (D
uring the talk I will explain the\nmeaning of these somewhat cryptic formu
lations.) Subsequently\, I\nwill show an application of the formula. I wil
l close the talk by\npresenting a generalisation that is inspired by Uvaro
v's formula\nfor the orthogonal polynomials of rationally related densitie
s.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shishuo Fu (Chongqing University)
DTSTART;VALUE=DATE-TIME:20210401T103000Z
DTEND;VALUE=DATE-TIME:20210401T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/21
DESCRIPTION:Title: Bijective recurrences for Schroeder triangles and Comtet statistics
\nby Shishuo Fu (Chongqing University) as part of Special Functions and Nu
mber Theory seminar\n\n\nAbstract\nIn this talk\, we bijectively establish
recurrence relations for two triangular arrays\, relying on their interpr
etations in terms of Schroeder paths (resp. little Schroeder paths) with g
iven length and number of hills. The row sums of these two triangles produ
ce the large (resp. little) Schroeder numbers. On the other hand\, it is w
ell-known that the large Schroeder numbers also enumerate separable permut
ations. This propelled us to reveal the connection with a lesser-known per
mutation statistic\, called initial ascending run (iar)\, whose distributi
on on separable permutations is shown to be given by the first triangle as
well. A by-product of this result is that "iar" is equidistributed over s
eparable permutations with "comp"\, the number of components of a permutat
ion. We call such statistics Comtet and we briefly mention further work co
ncerning Comtet statistics on various classes of pattern avoiding permutat
ions. The talk is based on joint work with Zhicong Lin and Yaling Wang.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nayandeep Deka Baruah (Tezpur University)
DTSTART;VALUE=DATE-TIME:20210415T103000Z
DTEND;VALUE=DATE-TIME:20210415T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/22
DESCRIPTION:Title: Matching coefficients in the series expansions of certain $q$-products
and their inverses\nby Nayandeep Deka Baruah (Tezpur University) as pa
rt of Special Functions and Number Theory seminar\n\n\nAbstract\nWe show t
hat the series expansions of certain $q$-products have \\textit{matching
coefficients} with their inverses. Several of the results are associated t
o Ramanujan's continued fractions. For example\, let $R(q)$ denote the Rog
ers-Ramanujan continued fraction having the well-known $q$-product repesen
tation $R(q)=\\left(q\,q^4\;q^5\\right)_{\\infty}/\\left(q^2\,q^3\;q^5\\ri
ght)_{\\infty}$. 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^{\\prim
e}(n)q^n\\right)^{-1}\,\\\\\n\\sum_{n=0}^{\\infty}\\beta(n)q^n=\\dfrac{R(q
)}{R\\left(q^{16}\\right)}=\\left(\\sum_{n=0}^{\\infty}\\beta^{\\prime}(n)
q^n\\right)^{-1}\,\n\\end{align*}\nthen\n\\begin{align*}\n\\alpha(5n+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://researchseminars.org/talk/SF-and-nt/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ankush Goswami (IIT Gandhinagar)
DTSTART;VALUE=DATE-TIME:20210527T103000Z
DTEND;VALUE=DATE-TIME:20210527T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/23
DESCRIPTION:Title: Partial theta series with periodic coefficients and quantum modular for
ms\nby Ankush Goswami (IIT Gandhinagar) as part of Special Functions a
nd Number Theory seminar\n\n\nAbstract\nTheta series first appeared in Eu
ler’s work on partitions\, but was systematically studied later by Jacob
i. In his Lost Notebook\, Ramanujan wrote down many identities (without p
roof) involving the so-called partial theta series. Unlike the theta serie
s which are modular forms\, the theory of partial theta series is not well
understood. In this talk\, I will consider a family of partial theta seri
es and show their “quantum modular” behaviour. This is based on my rec
ent joint work with Robert Osburn (UCD).\n\nThe talk should be accessible
to graduate and advanced undergraduate students.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritabrata Munshi (ISI\, Kolkata)
DTSTART;VALUE=DATE-TIME:20210610T103000Z
DTEND;VALUE=DATE-TIME:20210610T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/24
DESCRIPTION:Title: 100 years of sub-convexity\nby Ritabrata Munshi (ISI\, Kolkata) as
part of Special Functions and Number Theory seminar\n\n\nAbstract\nI will
present a historical survey of the sub-convexity problem.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Debika Banerjee (IIIT\, Delhi)
DTSTART;VALUE=DATE-TIME:20210624T103000Z
DTEND;VALUE=DATE-TIME:20210624T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/25
DESCRIPTION:Title: Bessel functions and their application to classical number theory\n
by Debika Banerjee (IIIT\, Delhi) as part of Special Functions and Number
Theory seminar\n\n\nAbstract\nFinding solutions of differential equations
has been a problem in pure mathematics since the invention of calculus by
Newton and Leibniz in the 17th century. Bessel functions are solutions of
a particular differential equation\, called Bessel’s equation. In class
ical analytic number theory\, there are several summation formulas or trac
e formulas involving Bessel functions. Two prominent such are the Kuznetso
v trace formula and the Voronoi summation formula. In this talk\, I will p
resent some Voronoi type summation formulas and its application to Number
theory.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bibekananda Maji (IIT\, Indore)
DTSTART;VALUE=DATE-TIME:20210708T103000Z
DTEND;VALUE=DATE-TIME:20210708T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/26
DESCRIPTION:Title: On Ramanujan's formula for $\\zeta(1/2)$ and $\\zeta(2m+1)$\nby Bib
ekananda Maji (IIT\, Indore) as part of Special Functions and Number Theor
y seminar\n\n\nAbstract\nEuler's remarkable formula for $\\zeta(2m)$ immed
iately tells us that even zeta values are transcendental. However\, the al
gebraic nature of odd zeta values is yet to be determined. \nPage 320 and
332 of Ramanujan's Lost Notebook contains an intriguing identity for $\\z
eta(2m+1)$ and $\\zeta(1/2)$\, respectively. Many mathematicians have stu
died these identities over the years.\n\nIn this talk\, we shall discuss t
ransformation formulas for a certain infinite series\, which will enable
us to derive Ramanujan's formula for $\\zeta(1/2)\,$ Wigert's formula 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 Ramanujan.\n\nT
his is joint work with Anushree Gupta.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anup Biswanath Dixit (IMSc (Chennai))
DTSTART;VALUE=DATE-TIME:20210722T103000Z
DTEND;VALUE=DATE-TIME:20210722T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/27
DESCRIPTION:Title: On Euler-Kronecker constants and the class number problem\nby Anup
Biswanath Dixit (IMSc (Chennai)) as part of Special Functions and Number T
heory seminar\n\n\nAbstract\nAs a natural generalization of the Euler's co
nstant \n$\\gamma$\, Y. Ihara introduced the Euler-Kronecker constants at
tached \nto any number field. In this talk\, we will discuss the connecti
on \nbetween these constants and certain arithmetic properties of number
\nfields.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter A Clarkson (University of Kent\, Canterbury\, UK)
DTSTART;VALUE=DATE-TIME:20210805T103000Z
DTEND;VALUE=DATE-TIME:20210805T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/28
DESCRIPTION:Title: Special polynomials associated with the Painlev\\'{e} equations\nby
Peter A Clarkson (University of Kent\, Canterbury\, UK) as part of Specia
l Functions and Number Theory seminar\n\n\nAbstract\nThe six Painlev\\'{e}
equations\, whose solutions are called the Painlev\\'{e} transcendents\,
were derived by Painlev\\'{e} and his colleagues in the late 19th and earl
y 20th centuries in a classification of second order ordinary differential
equations whose solutions have no movable critical points.\nIn the 18th a
nd 19th centuries\, the classical special functions such as Bessel\, Airy\
, Legendre and hypergeometric functions\, were recognized and developed in
response to the problems of the day in electromagnetism\, acoustics\, hyd
rodynamics\, elasticity and many other areas.\nAround the middle of the 20
th century\, as science and engineering continued to expand in new directi
ons\, a new class of functions\, the Painlev\\'{e} functions\, started to
appear in applications. The list of problems now known to be described by
the Painlev\\'{e} equations is large\, varied and expanding rapidly. The l
ist includes\, at one end\, the scattering of neutrons off heavy nuclei\,
and at the other\, the distribution of the zeros of the Riemann-zeta funct
ion on the critical line $\\mbox{Re}(z) =\\tfrac12$. Amongst many others\,
there is random matrix theory\, the asymptotic theory of orthogonal polyn
omials\, self-similar solutions of integrable equations\, combinatorial pr
oblems such as the longest increasing subsequence problem\, tiling problem
s\, multivariate statistics in the important asymptotic regime where the n
umber of variables and the number of samples are comparable and large\, an
d also random growth problems.\n\nThe Painlev\\'{e} equations possess a pl
ethora of interesting properties including a Hamiltonian structure and ass
ociated isomonodromy problems\, which express the Painlev\\'{e} equations
as the compatibility condition of two linear systems. Solutions of the Pai
nlev\\'{e} equations have some interesting asymptotics which are useful in
applications. They possess hierarchies of rational solutions and one-para
meter families of solutions expressible in terms of the classical special
functions\, for special values of the parameters. Further the Painlev\\'{e
} equations admit symmetries under affine Weyl groups which are related to
the associated B\\"ack\\-lund transformations.\n\nIn this talk I shall di
scuss special polynomials associated with rational solutions of Painlev\\'
{e} equations. Although the general solutions of the six Painlev\\'{e} equ
ations are transcendental\, all except the first Painlev\\'{e} equation po
ssess rational solutions for certain values of the parameters. These solut
ions are expressed in terms of special polynomials\nThe roots of these spe
cial polynomials are highly symmetric in the complex plane and speculated
to be of interest to number theorists. The polynomials arise in applicatio
ns such as random matrix theory\, vortex dynamics\, in supersymmetric quan
tum mechanics\, as coefficients of recurrence relations for semi-classical
orthogonal polynomials and are examples of exceptional orthogonal polynom
ials.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter A Clarkson (University of Kent\, Canterbury\, UK)
DTSTART;VALUE=DATE-TIME:20210805T103000Z
DTEND;VALUE=DATE-TIME:20210805T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/29
DESCRIPTION:Title: Special polynomials associated with the Painlev\\'{e} equations\nby
Peter A Clarkson (University of Kent\, Canterbury\, UK) as part of Specia
l Functions and Number Theory seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajat Gupta (IIT\, Gandhinagar)
DTSTART;VALUE=DATE-TIME:20210819T103000Z
DTEND;VALUE=DATE-TIME:20210819T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/30
DESCRIPTION:Title: Koshliakov zeta functions and modular relations\nby Rajat Gupta (II
T\, Gandhinagar) as part of Special Functions and Number Theory seminar\n\
n\nAbstract\nNikolai Sergeevich Koshliakov was an outstanding Russian math
ematician who made phenomenal contributions to number theory and different
ial equations. In the aftermath of World War II\, he was one among the man
y scientists who were arrested on fabricated charges and incarcerated. Und
er extreme hardships while still in prison\, Koshliakov (under a different
name `N. S. Sergeev') wrote two manuscripts out of which one was lost. Fo
rtunately the second one was published in 1949 although\, to the best of o
ur knowledge\, no one studied it until the last year when Prof. Atul Dixit
and I started examining it in detail. This manuscript contains a complete
theory of two interesting generalizations of the Riemann zeta function ha
ving their genesis in heat conduction and is truly a masterpiece! In this
talk\, we will discuss some of the contents of this manuscript and then pr
oceed to give some new results (modular relations) that we have obtained i
n this theory. This is joint work with Prof. Atul Dixit.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Howard Cohl (NIST)
DTSTART;VALUE=DATE-TIME:20210902T130000Z
DTEND;VALUE=DATE-TIME:20210902T140000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/31
DESCRIPTION:Title: The utility of integral representations for the Askey-Wilson polynomial
s and their symmetric sub-families\nby Howard Cohl (NIST) as part of S
pecial Functions and Number Theory seminar\n\n\nAbstract\nThe Askey-Wilson
polynomials are a class of orthogonal polynomials which are symmetric in
four free parameters which lie at the very top of the q-Askey scheme of ba
sic hypergeometric orthogonal polynomials. These polynomials\, and the pol
ynomials in their subfamilies\, are usually defined in terms of their fini
te series representations which are given in terms of terminating basic hy
pergeometric series. However\, they also have nonterminating\, q-integral\
, and integral representations. In this talk\, we will explore some of wha
t is known about the symmetry of these representations and how they have b
een used to compute their important properties such as generating functio
ns. This study led to an extension of interesting contour integral represe
ntations of sums of nonterminating basic hypergeometric functions initiall
y studied by Bailey\, Slater\, Askey\, Roy\, Gasper and Rahman. We will al
so discuss how these contour integrals are deeply connected to the propert
ies of the symmetric basic hypergeometric orthogonal polynomials.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neelam Saikia (University of Virginia)
DTSTART;VALUE=DATE-TIME:20210916T103000Z
DTEND;VALUE=DATE-TIME:20210916T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/32
DESCRIPTION:Title: Frobenius trace distributions for Gaussian hypergeometric functions
\nby Neelam Saikia (University of Virginia) as part of Special Functions a
nd Number Theory seminar\n\n\nAbstract\nIn the 1980's\, Greene defined hyp
ergeometric functions over finite fields using Jacobi sums. These function
s possess many properties that are analogous to those of the classical hyp
ergeometric series studied by Gauss\, Kummer and others. These functions h
ave played important roles in the study of supercongruences\, the Eichler-
Selberg trace formula\, and zeta-functions of arithmetic varieties. We stu
dy the distribution (over large finite fields) of the values of certain fa
milies of these functions. For the $_2F_1$ functions\, the limiting distri
bution is semicircular\, whereas the distribution for the $_3F_2$ function
s is the more exotic \\it{Batman distribution.}\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jaban Meher (NISER\, Bhubaneswar)
DTSTART;VALUE=DATE-TIME:20210930T103000Z
DTEND;VALUE=DATE-TIME:20210930T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/33
DESCRIPTION:Title: Modular forms and certain congruences\nby Jaban Meher (NISER\, Bhub
aneswar) as part of Special Functions and Number Theory seminar\n\n\nAbstr
act\nIn this talk we shall discuss about modular forms and certain types o
f congruences among the Fourier coefficients of modular forms. We shall al
so discuss about the non-existence of Ramanujan-type congruences for certa
in modular forms.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Meesue Yoo (Chungbuk National University\, Korea.)
DTSTART;VALUE=DATE-TIME:20211014T103000Z
DTEND;VALUE=DATE-TIME:20211014T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/34
DESCRIPTION:Title: Elliptic rook and file numbers\nby Meesue Yoo (Chungbuk National Un
iversity\, Korea.) as part of Special Functions and Number Theory seminar\
n\n\nAbstract\nIn this talk\, we construct elliptic analogues of the rook
numbers and file numbers by attaching elliptic weights to the cells in a b
oard. We show that our elliptic rook and file numbers satisfy elliptic ext
ensions of corresponding factorization theorems which in the classical cas
e was established by Goldman\, Joichi and White and by Garsia and Remmel i
n the file number case. This factorization theorem can be used to define e
lliptic analogues of various kinds of Stirling numbers of the first and se
cond kind\, and Abel numbers. \n\n\nWe also give analogous results for mat
chings of graphs\, elliptically extending the result of Haglund and Remmel
.\n\n\nThis is joint work with Michael Schlosser.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Akshaa Vatwani (IIT\, Gandhinagar)
DTSTART;VALUE=DATE-TIME:20211028T103000Z
DTEND;VALUE=DATE-TIME:20211028T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/35
DESCRIPTION:Title: Limitations to equidistribution in arithmetic progressions\nby Aksh
aa Vatwani (IIT\, Gandhinagar) as part of Special Functions and Number The
ory seminar\n\n\nAbstract\nIt is well known that the prime numbers are equ
idistributed in arithmetic progressions. Such a phenomenon is also observe
d more generally for a class of arithmetic functions. A key result in this
context is the Bombieri-Vinogradov theorem which establishes that the pri
mes are equidistributed in arithmetic progressions ``on average" for modul
i $q$ in the range $q \\le x^{1/2 -\\epsilon }$ for any $\\epsilon>0$. In
1989\, building on an idea of Maier\, Friedlander and Granville showed tha
t such equidistribution results fail if the range of the moduli $q$ is ext
ended to $q \\le x/ (\\log x)^B $ for any $B>1$. We discuss variants of th
is result and give some applications. This is joint work with Aditi Savali
a.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kamalakshya Mahatab (Chennai Mathematical Institute (CMI))
DTSTART;VALUE=DATE-TIME:20211111T103000Z
DTEND;VALUE=DATE-TIME:20211111T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/36
DESCRIPTION:Title: Large oscillations of the argument of the Riemann zeta function\nby
Kamalakshya Mahatab (Chennai Mathematical Institute (CMI)) as part of Spe
cial Functions and Number Theory seminar\n\n\nAbstract\nWe will obtain lar
ge values of the argument of the Riemann zeta function using the resonance
method. We will also apply the method to the iterated arguments. This is
a joint work with A. Chirre\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Koustav Banerjee (RISC\, JKU\, Linz)
DTSTART;VALUE=DATE-TIME:20211125T103000Z
DTEND;VALUE=DATE-TIME:20211125T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/37
DESCRIPTION:Title: Inequalities for the modified Bessel function of first kind and its con
sequences\nby Koustav Banerjee (RISC\, JKU\, Linz) as part of Special
Functions and Number Theory seminar\n\n\nAbstract\nStudy on asymptotics of
modified Bessel functions dates back to 18th century.\nIn this talk\, we
will describe how from the study of asymptotics of modified Bessel functio
n of\nfirst kind of non-negative order\, one can comes up with a host of i
nequalities that finally leads\nto answer combinatorial properties\, for e
xample log-concavity\, higher order Tur\\acute{a}n inequality\nof certain
arithmetic sequences arising from Fourier coefficients of modular forms. I
n addition\nto that\, we will discuss briefly on a result of Bringmann et
al. and analyze with the work\naddressed above.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Sokal (University College\, London and NYU)
DTSTART;VALUE=DATE-TIME:20030127T103000Z
DTEND;VALUE=DATE-TIME:20030127T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/38
DESCRIPTION:Title: Coefficientwise Hankel-total positivity\nby Alan Sokal (University
College\, London and NYU) as part of Special Functions and Number Theory s
eminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Sokal (University College\, London and NYU)
DTSTART;VALUE=DATE-TIME:20220127T103000Z
DTEND;VALUE=DATE-TIME:20220127T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/39
DESCRIPTION:Title: Coefficientwise Hankel-total positivity\nby Alan Sokal (University
College\, London and NYU) as part of Special Functions and Number Theory s
eminar\n\n\nAbstract\nA matrix $M$ of real numbers is called {\\em totally
positive}\\/\n if every minor of $M$ is nonnegative. Gantmakher and Kr
ein 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 underlying\n
sequence $(a_n)_{n \\ge 0}$ is a Stieltjes moment sequence.\n Moreover\,
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 continued frac
tion 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 the sens
e of formal power series) with all $\\alpha_i \\ge 0$.\n So totally posi
tive Hankel matrices are closely connected with\n the Stieltjes moment p
roblem and with continued fractions.\n\n Here I will introduce a general
ization: a matrix $M$ of polynomials\n (in some set of indeterminates)
will be called\n {\\em coefficientwise totally positive}\\/ if every min
or of $M$\n is a polynomial with nonnegative coefficients. And a seque
nce\n $(a_n)_{n \\ge 0}$ of polynomials will be called\n {\\em coeffic
ientwise Hankel-totally positive}\\/ if the Hankel matrix\n $H = (a_{i+j
})_{i\,j \\ge 0}$ associated to $(a_n)$ is coefficientwise\n totally po
sitive. It turns out that many sequences of polynomials\n arising natur
ally in enumerative combinatorics are (empirically)\n coefficientwise Ha
nkel-totally positive. In some cases this can\n be proven using continu
ed fractions\, by either combinatorial or\n algebraic methods\; I will
sketch how this is done. In many other\n cases it remains an open probl
em.\n\n One of the more recent advances in this research is perhaps of\n
independent interest to special-functions workers:\n we have found br
anched continued fractions for ratios of contiguous\n hypergeometric ser
ies ${}_r \\! F_s$ for arbitrary $r$ and $s$\,\n which generalize Gauss'
continued fraction for ratios of contiguous\n ${}_2 \\! F_1$. For the
cases $s=0$ we can use these to prove\n coefficientwise Hankel-total pos
itivity.\n\n Reference: Mathias P\\'etr\\'eolle\, Alan D.~Sokal and Bao-
Xuan Zhu\,\n arXiv:1807.03271\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Surbhi Rai (IIT\, Delhi)
DTSTART;VALUE=DATE-TIME:20220210T103000Z
DTEND;VALUE=DATE-TIME:20220210T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/40
DESCRIPTION:Title: Expansion Formulas for Multiple Basic Hypergeometric Series Over Root
Systems\nby Surbhi Rai (IIT\, Delhi) as part of Special Functions and
Number Theory seminar\n\n\nAbstract\nIn a series of works\, Zhi-Guo Liu ex
tended some of the central summation and transformation formulas of basic
hypergeometric series. In particular\, Liu extended Rogers' non-terminatin
g very-well-poised $_{6}\\phi_{5}$ summation formula\, Watson's transfor
mation\nformula\, and gave an alternate approach to the orthogonality of t
he Askey-Wilson polynomials. These results are helpful in number-theoretic
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 f
undamental formulas to multiple basic hypergeometric series over root syst
ems. We will also discuss results that extend Wang and Ma's generalization
s of Liu's work which they obtained using $q$-Lagrange inversion. Subseque
ntly\, we will look at an application based on the expansions of infinite
products. These extensions have been obtained using the $A_n$ and $C_n$ Ba
iley transformation and other summation theorems due to Gustafson\, Milne\
, Milne and Lilly\, and others\, from $A_n$\, $C_n$ and $D_n$ basic hyperg
eometric series theory. We will observe how this approach brings Liu's exp
ansion formulas within the Bailey transform methodology.\n\nThis talk is b
ased on joint work with Gaurav Bhatnagar. (https://arxiv.org/abs/2109.0282
7)\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krishnan Rajkumar (JNU)
DTSTART;VALUE=DATE-TIME:20220224T103000Z
DTEND;VALUE=DATE-TIME:20220224T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/41
DESCRIPTION:Title: The Binet function and telescoping continued fractions\nby Krishnan
Rajkumar (JNU) as part of Special Functions and Number Theory seminar\n\n
\nAbstract\nThe Binet function $J(z)$ defined by the equation $\\Gamma(z)
= \\sqrt{2 \\pi} z^{z-\\frac{1}{2}}e^{-z} e^{J(z)}$ is a well-studied fun
ction. The Stirling approximation comes from the property $J(z) \\rightarr
ow 0$ as $z\\rightarrow \\infty$\, $|arg z|<\\pi$. In fact\, an asymptotic
expansion $J(z) \\sim z^{-1} \\sum_{k=0}^{\\infty} c_k z^{-2k}$ holds in
this region\, with closed form expressions for $c_k$ and explicit integral
s for the error term for any finite truncation of this asymptotic series.
\n\nIn this talk\, we will discuss two different classical directions of r
esearch. 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 e
stablish inequalities of the form $J(n)-J(n+1) > F(n)-F(n+1)$ for an appro
priate rational function $F$ to derive the corresponding lower bounds by t
elescoping.\n\nThe second direction is to use moment theory to derive cont
inued fractions of specified forms for $J(x)$. For instance\, a modified S
-fraction of the form $\\frac{a_1}{x \\ +} \\frac{a_2}{x \\ +}\\frac{a_3}{
x \\ +} \\cdots$ can be formally derived from the above asymptotic expansi
on using a method called the qd-algorithm. The resulting continued fractio
n can then be shown to converge to $J(x)$ by the asymptotic properties of
$c_k$ and powerful results from moment theory. There are no known closed-f
orm expressions for the $a_k$.\n\nWe will then outline what we call the me
thod of telescoping continued fractions to extend the elementary methods o
f the first approach to derive the modified S-fraction for $J(x)$ obtained
in the second by a new algorithm. We will describe several results that w
e can prove and some conjectures that together enhance our understanding o
f the numbers $a_k$ as well as provide upper and lower bounds for $J(x)$ t
hat improve all known results.\n\nThis is joint work with Gaurav Bhatnagar
.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenguang Zhai (China Institute of Mining and Technology\, Beijing\
, PRC)
DTSTART;VALUE=DATE-TIME:20220310T103000Z
DTEND;VALUE=DATE-TIME:20220310T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/42
DESCRIPTION:Title: The MC-algorithm and continued fraction formulas involving ratios of Ga
mma functions\nby Wenguang Zhai (China Institute of Mining and Technol
ogy\, Beijing\, PRC) as part of Special Functions and Number Theory semina
r\n\n\nAbstract\nRamanujan discovered many continued fraction expansions a
bout ratios of the Gamma functions. However\, Ramanujan left us no clues a
bout how he discovered these elegant formulas. In this talk\, we will expl
ain the so-called MC-algorithm. By this algorithm\, we can not only redisc
over many of Ramanujan's continued fraction expansions\, but also find som
e new formulas.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumyarup Banerjee (IIT\, Gandhinagar)
DTSTART;VALUE=DATE-TIME:20220324T103000Z
DTEND;VALUE=DATE-TIME:20220324T113000Z
DTSTAMP;VALUE=DATE-TIME:20230921T144555Z
UID:SF-and-nt/43
DESCRIPTION:Title: Finiteness theorems with almost prime inputs\nby Soumyarup Banerjee
(IIT\, Gandhinagar) as part of Special Functions and Number Theory semina
r\n\n\nAbstract\nThe Conway–Schneeberger Fifteen theorem states that a g
iven positive definite integral quadratic form is universal (i.e.\, repres
ents every positive integer with integer inputs) if and only if it represe
nts the integers up to 15. This theorem is sometimes known as “Finitenes
s 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 i
nvestigated quadratic forms which are universal when restricted to almost
prime inputs and have established finiteness theorems akin to the Conway
–Schneeberger Fifteen theorem.\n
LOCATION:https://researchseminars.org/talk/SF-and-nt/43/
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