The Online
Encyclopedia Riordanica
Prof.Em. Giacomo Della Riccia
Dept. of Mathematics, Informatics and Physics
(DIMIF)
– Research Center Norbert Wiener
University of Udine
Via delle Scienze 206-33100
Udine (Italy)
dlrca@uniud.it
Preface
Triangular
arrays and general Riordan arrays are
studied in J. O. Shallit (Univ.
of
Waterloo) 1980 paper "A triangle for the Bell numbers"
and A. Nkwanta (Morgan
State Univ.) articles, cited below. For the History, we recall that
these are
the starting points of the “Online Encyclopedia Riordanica (OERIOR)”
with the purpose to encourage research on topics related to Riordan
arrays/Riodan group,
to
provide assistance in the preparation of a
thesis, to stimulate
graduate students and researchers who want to get more insight on a
specific
topic, to provide References and Citations for new Publications.
In
2014, OERIOR included only about hundred
Publications. I sent this material to A. Nkwanta
and
G-S. Cheon (Sungkyunkwan Univ.),
with a kind request to express their opinion on the project. Their prompt
and enthousiastic reply
encouraged me to continue the “Online Encyclopedia Riordanica (OERIOR)”.
1.-
Introduction
Oerior is
articulated in 3 parts: Database, Glossary,
Bibliography. Database is
a survey of articles relevant to Oerior, Glossary is
a survey of labelled Directories
and Bibliography is a list of
recommended readings. The
Content of each part is accessible by clicking on the corresponding
Link (see
below). Citations are written in black if
a free
copy is available and/or an Open Access policy is applicable and
in red if
only an Abstract is available due to Purchase requests. The articles in
the Database are
ordered by the authors family names and date
of
publication; when papers by the same authors appear
the same year, we
also use letters a, b, c, etc....after the year, as in the following
examples:
Azarian2012a, Fibonacci
identities as binomial
sums, Int. J. Contemp. Math. Sci. Vol. 7, 2012, no.
38,
1871-1876, gen>
Azarian2012b, Fibonacci
identities as binomial
sums II, Int. J. Contemp. Math. Sci. Vol. 7, 2012,
no. 42,
2053-2059, gen>
Azarian2012c, Identities
involving Lucas or
Fibonacci and Lucas numbers as binomial sums, Int.
J. Contemp. Math.
Sci. Vol. 7, 2012, no. 45, 2221-2227, gen>
Cheon G-S.2003, A
note on the Bernoulli and Euler polynomials, Appl.
Math. Letters Vol.
16, Issue 3, Apr 2003, 365–368, gen>
Cheon G-S.El-Mikkawy2007, Generalized
harmonic numbers identities and a related matrix
representation, J.
Korean Math. Soc. 2007 Vol. 44, No. 2, 487-498, nat>
Cheon S.El-Mikkawy2008, Generalized
harmonic numbers with Riordan arrays, J. Number
Theory Vol. 128, Issue
2, Feb 2008, 413–425, jou>
Cheon G-S.
HwangRimSong2003, Matrices determined by a linear
recurrence relation
among entries, Linear Algebra Appl Vol.
373,
Nov2003, 89–99, gen>
Cheon G-S.Jin2011, Structural
properties of Riordan matrices and extending the matrices, Linear
Algebra Appl Vol.
435,
Issue 8, Oct 2011, 2019–2032, gen>
Cheon G-S.JinKimShapiro2009, Riordan
group involutions and the Δ-sequence, Discrete
Appl. Math. 157
(2009) 1696-1701, gen>
Cheon G-S.Kim2001, Stirling matrix
via
Pascal matrix, Linear
Algebra
Appl. Vol. 329, Issues 1–3, May 2001,
49–59, gen>
Cheon G-S.Kim2002, Factorial Stirling matrix
and
related combinatorial sequences, Linear
Algebra
Appl. Vol. 357, Issues 1–3, Dec 2002,
247–258, gen>
Cheon G-S.Kim2008, Simple
proofs of open problems about the structure of involutions in the
Riordan
group, Linear Algebra Appl. Vol. 428, Issue 4, Feb
2008,
930–940, gen>
Cheon G-S.KimShapiro2008, Riordan
group involutions, Linear Algebra Appl. Vol. 428,
Issue 4, Feb 2008,
941–952, gen>
Cheon G-S.KimShapiro2009, A
generalization of Lucas polynomial sequence, Discrete
Appl. Math. Vol.
157, Issue 5, Mar 2009, 920–927, gen>
Cheon G-S.KimShapiro2012, Combinatorics of
Riordan
arrays with identical A and Z sequences, Discrete
Math.
Vol. 312, Issues 12–13, Jul 2012,
2040–2049, gen>
Cheon G-S.YungLim2013, A
q-analogue of the Riordan group, Linear
Algebra Appl Vol.
439,
Issue 12, Dec 2013, 4119–4129, gen>
Nkwanta2003, A
Riordan matrix approach to unifying a selected class of combinatorial
arrays, Congr. Numer.
160 (2003),
33-45, gen>
Nkwanta2008, Lattice
Paths, Riordan Matrices and RNA Numbers, Congr. Numer. 01/2008, gen>
Nkwanta2009, Lattice path and RNA secondary structure predictions, 15th Conf. African American Researchers Math. Sci.-Rice Univ., Jun 23-26, 2009, gen>
Nkwanta2010, Riordan matrices and higher-dimensional lattice walks, J. of Statist. Plann. Inference Vol. 140, Issue 8, Aug 2010, 2321–2334, jou>
NkwantaBarnes2012, Two
Catalan-type Riordan arrays and their connections to the Chebyshev polynomials
of the first kind, J.
Integer Seq.
Vol. 15 (2012), Article 12.3.3, jis>
NkwantaKnox1999, A
note on Riordan matrices, Thesis-Contemp. Math. Vol.
252. 1999, Howard
University, Washington, DC 1997, gen>
NkwantaShapiro2005, Pell
walks and Riordan matrices, Fibonacci Quart. 2005
(43,2):
170-180, fibqy>
NkwantaTefera2013, Curious
relations and identities involving the Catalan generating function and
numbers, J. of Integer Seq. Vol. 16 (2013), Article
13.9.5, jis>
Page
1
Glossary-Keywords (to
see the details of Glossary-Keyword, CTRL
and click here . Glossary-Keywords)
Abel
Akiyama-Tanigawa
Al-Salam-Carlitz
Al-Salam-Chihara
Apery
Apostol
Apostol-Bernoulli
Apostol-Euler
Apostol-Genocchi
Appel
array type polynomials
Askey scheme
Askey-Wilson algebra
Askey-Wilson
Barnes-type
basis
Bell
Bell partial polynomials
Bernoulli
Bernstein
Bessel big q-analogues
Bessel
Binet formula
binomial
Brownian motion, Brownian motion q-analogue
Carlitz
Catalan
Cauchy
central coefficients
central factorial numbers
Chan-Chyan-Srivastava
Charlier
Chebyshev(Tschebyscheff)
Chebyshev-Boubaker
circulant matrices
coefficients method
Cohen-Macaulay property
combinatorial theory
Comtet
congruences
connection coefficients
continued fractions
convolution
cumulants
Daehee
degenerate numbers, degenerate
polynomials
Delannoy
Denert statistic
derangements, derangements q-analogues
Diophantine equations
Dobinski
Dumont-Foata
Ehrhart
elliptic (see also Jacobi)
embedding distributions, structures
Entriger
entropy
Erkus-Srivastava
Euler
Euler-Barnes
Euler-Bernoulli
Euler-Frobenius
Eulerian
Euler-Seidel
Faber
factorial generalizations (q-)numbers,
(q-)polynomials
Fibonacci
Fibonacci-Lucas
Fibonomial coefficients
Fine
Frobenius
Gandhi
Gauss
(see also hypergeometric)
Gegenbauer (see also ultraspherical)
Gegenbauer-Humbert
generating functions
Genocchi
Hahn
Hahn's theorem
Hankel
harmonic
Hermite
Hermite big q-polynomials
Hessenberg
Horadam
Humbert
hypergeometric (see also Gauss)
identities, inequalities
incomplete numbers, generalized numbers,
polynomials
integer sequences
inverse
(reciprocal) numbers, sums, polynomials
inversion techniques
Jacobi big q-polynomials
Jacobi little q-polynomials
Jacobi (see also elliptic)
Jacobi-Stirling
Jacobsthal
Jacobsthal-Lucas
Konhauser
Krawtchouk
lacunary series
Lagrange
Laguerre little q-polynomials
Laguerre
Lah
lattice
Laurent
LDU decomposition, Cholesky factorization
Legendre
Legendre-Stirling
Lehmer
Lehner
Lengyel
L-functions
linear algebra of certain matrices
Lucas
Lucas-Bernoulli
Lucasian
Mahonian pairs, statistics
Meixner
Mellin
ménage problem
mixed-type polynomials
modular
moments
Morgan-Voyce
Motzkin
Narayana
Narumi
n-bonacci numbers
Newton series
Norlund
Norlund-Bernoulli
Norlund-Euler
operational calculus
Oresme
orthogonal (q-)polynomials
partial Euler product
Pascal
paths
patterns
Pell
Pell equation, Pell-Abel equation
Pell-Lucas
permanents
permutations
Perrin
Poisson-Charlier
poly-numbers, poly-polynomials
posets
process
production matrices
q-analogue calculus
Racah coefficients
recurrence relations
renewal array, process
Riemann (see also z-function)
Riordan group (q-)analogue
RNA secondary structures, numbers
Rodrigues
Salié
Schröder
Schubert
Schur
Seidel-Arnold
Selberg
Sheffer group
Sheffer polynomial sequences
Sheffer-type
Sobolev
Somos-4 sequences
Springer
Srivastava
Srivastava-Pintér addition theorems
Stern-Brocot sequence
Stieltjes
Stirling
Stirling generalized numbers group
stochastic processes
succession rules
Sulanke
tangent numbers, tanh numbers
Tetranacci
Toda chain
Toeplitz
Toeplitz plus Hankel matrices
Touchard
transforms
Tribonacci
Tribonacci-Lucas
ultraspherical (see also Gegenbauer)
Umbral calculus
van der Laan
Vandermonde
Vieta, Vieta-Pell,
Vieta-Pell_Luca polynomials
Vieta-Jacobsthal-Lucas,
Vieta-Pell_Lucas polynomials
Weierstrass
Wiener chaos
Wythoff number, pair
Zernike
z-function (see also Riemann)
Page2
Meixner
BozejkoDemni2010, Topics on Meixner families, Banach Center Publications,
2010 Vol. 89, 61-74, nat>
Meixner-Riordan
arrays
BarryHennessy2010b, Meixner-type results for Riordan arrays and associated integer sequences, J. Integer Seq.
Vol. 13 (2010), Article 10.9.4, jis>
Meixner-type
BarryHennessy2010b, Meixner-type results for Riordan arrays and associated integer sequences, J. Integer Seq.
Vol. 13 (2010), Article 10.9.4, jis>
Meixner polynomials
Alvarez-NodarseMarcellan1995b, Difference equation for modifications of Meixner polynomials, J. Math. Anal. Appl. Vol. 194, Issue 1, Aug 1995,
250–258, jou>
Bavinck, van Haeringen1994, Difference equations for generalized Meixner polynomials, J. Math. Anal. Appl. Vol. 184, Issue 3, Jun 1994,
453–463, jou>
BrycWesolowski2004, Conditional moments of q-Meixner processes, arXiv (13 Dec 2004), aXv>
GriffithsSpano2011, Multiv. Jacobi and Laguerre polyn., infinite-dimens. extensions and their prob. connect. with multiv. Hahn and Meixner polynomials, Bernoulli 17 (3),
2011, 1095–1125, gen>
KhanAkhlaq2012, A
note on generating functions and summation formulas for Meixner polynomials of several variables, Demonstratio Math. Vol. XLV, No. 1,
2012, gen>
Shibukawa2014, Multivariate Meixner, Charlier and Krawtchouk polynomials, arXiv (29 Apr 2014), aXv>
generating
functions
KhanAkhlaq2012, A
note on generating functions and summation formulas for Meixner polynomials of several variables, Demonstratio Math. Vol. XLV, No. 1, 2012, gen>
Hahn
GriffithsSpano2011, Multiv. Jacobi and Laguerre polyn., infinite-dimens. extensions and their prob. connect. with multiv. Hahn and Meixner polynomials, Bernoulli 17 (3), 2011,
1095–1125, gen>
integer
sequences
BarryHennessy2010b, Meixner-type results for
Riordan arrays and associated integer sequences, J. Integer
Seq. Vol. 13 (2010), Article 10.9.4, jis>
Jacobi (see also elliptic)
GriffithsSpano2011, Multiv. Jacobi and Laguerre polyn., infinite-dimens. extensions and their prob. connect. with multiv. Hahn and Meixner polynomials, Bernoulli 17 (3),
2011,
1095–1125, gen>
GriffithsSpano2011, Multiv. Jacobi and Laguerre polyn., infinite-dimens. extensions and their prob. connect. with multiv. Hahn and Meixner polynomials, Bernoulli 17 (3),
2011, 1095–1125, gen>
moments
BrycWesolowski2004, Conditional moments of q-Meixner processes, arXiv (13 Dec 2004), aXv>
process
BrycWesolowski2004, Conditional moments of q-Meixner processes, arXiv (13 Dec 2004), aXv>
The above
display is a so-called thematic map; more precisely
we say that it is
the Meixner Directory thematic map.
Usually a thematic map
is related to several keywords, in our case: generating
functions, Hahn,
integer sequences, Jacobi (see
also elliptic),
Krawtchouk, Laguerre, moments and process.
Displaying the
additional thematic maps, we get 9 thematic
maps which provide a
more detailed panoramic
view of
the topic. To save space, we have not displayed these thematic
maps. Thematic
maps are used in conjunction with the
applications mentioned
above; they represent an important
feature of OERIOR.
Page 2
Database
(to see the publications listed in the
Database, CTRL
and click here Database
).
Glossary-Database.docx, Glossary-Database.html, Glossary-Database.pdf.
Glossary-Keywords (to
see
the details of Glossary-Keyword, CTRL and click
here Glossary-Keywords).
Glossary
(to
see the details of Glossary, CTRL and click here Contents).Glossary-Contents.docx,
Glossary-Contents.html,
Glossary-Contents.pdf.Bibliography (to
see the items in the Bibliography, CTRL
and click
here Bibliography
).Glossary-Bibliography.docx,
Glossary-Bibliography.html,
Glossary-Bibliography.pdf
Items in OERIOR can be read on-line (CTRL and one: jis>, aXv>, gen>, jou>, nat>, fibqy (acronyms of Journal Integer Sequences, aXv, General, Journal, Fibquarterly,National). This original feature of OERIOR gives immediate access to a huge amount of information.
There
are 1959 entries in Database, 192
in Glossary and 79 in Bibliography .
These
numbers grow as new items are discovered in the literature due to
reader ’
contributions. Readers are welcome to send via email
suggestions for
further additions.
Database shows
that
only few items contain in their title the keywords Riordan
arrays/group; all
the others belong to OERIOR because they are included in a Directory
related
(CTRL and click here Contents)
to a Directory indexed by Riordan arrays/group.
OERIOR
is open/free and may be copied for personal
reading. We kindly ask users to publicize OERIOR by
including in
their publications the Reference
“G. DellaRiccia, Online
Encyclopedia Riordanica”, the Citation
"Online Encyclopedia Riordanica (Oerior)”
and the Link
I acknowledge with pleasure the excellent work
of
A. Angelucci (Univ. of L'Aquila) on
the Oerior webdesign,
the insertion by P. Corvaja (Univ. of Udine)
of the
"Jacobi (elliptic)" and "elliptic" entries in the Glossary and
his papers in the Database, the remarkable
work of V. Roberto
(Univ. of Udine) in the editing of Oerior,
the illstration by
R. Angeletti of the art of programming, the
presentation by G.L.
Franco (Dimif) and C. Maltese (Dimif) and, last but not
least, M. Di
Sabatini
for some software design
procedures.
Giacomo Della Riccia
(May 2017)