The Online Encyclopedia Riordanica

Prof.EmGiacomo 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 OeriorGlossary 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, CongrNumer. 160 (2003), 33-45, gen>

Nkwanta2008, Lattice Paths, Riordan Matrices and RNA Numbers, CongrNumer. 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
i
nverse (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 marices
Lucas
Lucas-Bernoulli
Lucasian
Mahonian pairs, statistics
Meixner
Mellin
menage 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
Rodrighes
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. MathAnalAppl. Vol. 194, Issue 1, Aug 1995, 250–258, jou>

Bavinckvan Haeringen1994, Difference equations for generalized Meixner polynomials, J. MathAnalAppl. Vol. 184, Issue 3, Jun 1994, 453–463, jou>

BrycWesolowski2004, Conditional moments of q-Meixner processesarXiv (13 Dec 2004), aXv>

GriffithsSpano2011, MultivJacobi and Laguerre polyn., infinite-dimensextensions and their probconnectwith multivHahn and Meixner polynomials, Bernoulli 17 (3), 2011, 1095–1125, gen>

KhanAkhlaq2012, A note on generating functions and summation formulas for Meixner polynomials of several variablesDemonstratio Math. Vol. XLV, No. 1, 2012, gen>

Shibukawa2014, Multivariate MeixnerCharlier and Krawtchouk polynomials, arXiv (29 Apr 2014), aXv>

generating functions

 KhanAkhlaq2012, A note on generating functions and summation formulas for Meixner polynomials of several variablesDemonstratio Math. Vol. XLV, No. 1, 2012, gen>

Hahn

GriffithsSpano2011, MultivJacobi and Laguerre polyn., infinite-dimensextensions and their probconnectwith multivHahn 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, MultivJacobi and Laguerre polyn., infinite-dimensextensions and their probconnectwith multivHahn and Meixner polynomials, Bernoulli 17 (3), 2011, 1095–1125, gen>

 

Laguerre

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 processesarXiv (13 Dec 2004), aXv>

 

process

BrycWesolowski2004, Conditional moments of q-Meixner processesarXiv (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



Conclusion

 

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

http://sole.dimi.uniud.it/~giacomo.dellariccia/onlinhttp://sole.dimi.uniud.it/~giacomo.dellariccia/online encyclopedia riordanica.html

 

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)