On the higher-order sheffer orthogonal polynomial sequences

118 Pages·2013·0.766 MB·English

by  Daniel Joseph Galiffa

SpringerBriefs in Mathematics
SeriesEditors
KrishnaswamiAlladi
NicolaBellomo
MicheleBenzi
TatsienLi
MatthiasNeufang
OtmarScherzer
DierkSchleicher
BenjaminSteinberg
VladasSidoravicius
YuriTschinkel
LoringW.Tu
G.GeorgeYin
PingZhang
SpringerBriefs in Mathematics showcases expositions in all areas of
mathematics and applied mathematics. Manuscripts presenting new results
or a single new result in a classical field, new field, or an emerging topic,
applications, or bridges between new results and already published works,
are encouraged. The series is intended for mathematicians and applied
mathematicians.
Forfurthervolumes:
http://www.springer.com/series/10030

Daniel Joseph Galiffa
On the Higher-Order Sheffer
Orthogonal Polynomial
Sequences
123
DanielJosephGaliffa
PennStateErie
TheBehrendCollege
Erie,Pennsylvania
USA
ISSN2191-8198 ISSN2191-8201(electronic)
ISBN978-1-4614-5968-2 ISBN978-1-4614-5969-9(eBook)
DOI10.1007/978-1-4614-5969-9
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012951348
MathematicsSubjectClassification(2010):33C45,33C47
©DanielJ.Galiffa2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Thismonographisdedicated tomynephew,
ShadanDaniel

Preface
In 1939, I.M. Sheffer published seminal results regarding the characterizations
of polynomials via general degree-lowering operators and showed that every
polynomial sequence can be classified as belonging to exactly one Type. A large
portion of his work was dedicated to developing a wealth of aesthetic results
regarding the most basic type set, entitled B-Type 0 (or equivalently A-Type 0),
whichincludedthe developmentof severalinteresting characterizingtheorems.In
particular, one of Sheffer’s most important results was his classification of which
B-Type 0 sets were also orthogonal, which are now known to be the very well-
studied and applicable Laguerre, Hermite, Charlier, Meixner, Meixner–Pollaczek,
andKrawtchoukpolynomials,whichare often called the ShefferSequences.Asit
turnedout,ShefferprovedthateveryB-Type0set{P(x)}∞ canbecharacterized
n n=0
bythegeneratingfunction:
∞
A(t)exH(t)= ∑P(x)tn,
n
n=0
where A(t) and H(t) are formal power series in t, with certain restrictions.
Furthermore, Sheffer also briefly described how this generating function can also
beextendedtothecaseofarbitraryB-Typekasfollows:
(cid:2) (cid:3) ∞
A(t)exp xH1(t)+···+xk+1Hk+1(t) = ∑Pn(x)tn,
n=0
with Hi(t)=hi,iti+hi,i+1ti+1+···, h1,1(cid:2)=0, i=1,2,...,k+1.
Thusfar,averylargeamountofresearchhasbeencompletedregardingthetheory
andapplicationsoftheB-Type0sets.Therefore,itisnaturaltoattempttodetermine
whether or not orthogonal sets can be extracted from the higher-order classes,
i.e., k≥1 in the generating relation directly above. In fact, no results have been
published to date that specifically analyze the higher-order Sheffer classes. With
this in mind, we have constructed the novel results of this monograph (Chap.3),
vii
viii Preface
whereinwepresentapreliminaryanalysisofaspecialcaseoftheB-Type1class.We
conductthisanalysisforthefollowingreasons.One,mostimportantly,ourmethod
functions as a template, which can be applied to other characterization problems
as well. In order to effectivelyapply this method, computer algebra was found to
be essential and Mathematica(cid:2)R was determined to be the most efficient platform
forperformingeachofourmanipulations.Therefore,oursecondmotivationisthe
fact that the novel analysis of Chap.3 lends itself as a paradigm regarding how
computer algebra packages, like Mathematica(cid:2)R, can play an important role in
developingrigorousmathematics.Lastly,weintendforthisworktoeventuallylead
to a complete characterization of the general B-Type k orthogonal sets and foster
futureresearchonothertypesofsimilarcharacterizationproblemsaswell.
We certainly wish to emphasize that Mathematica(cid:2)R was utilized only for
managingmanyofthealgebraicmanipulationsinvolvedinestablishingtheoriginal
resultswithin.TherelationshipsachievedwiththeaidofMathematica(cid:2)R wereused
to rigorously construct the novel theorems of this work, which were proven via
algebraictechniquesandrudimentarylinearalgebra,withouttheusageofcomputer
algebra.
Now, it is well known that symbolic computationsare becomingutilized more
frequently in mathematical research and are also becoming increasingly more
accepted.Severaljournalsincludevariousresultsthatarebasedoncomputeralgebra
andsomemayeveninclude,essentially,diminutivefragmentsofcode,pseudo-code,
orcomputeralgebraoutputs.InChap.3ofthiswork,awealthoftheMathematica(cid:2)R
inputs and their respective outputs are displayed in a reader-friendly format and
written in a distinctive font class that is similar to the Mathematica(cid:2)R notebook.
This amount of displayed code is certainly not typical in any of the current peer-
reviewed mathematical science journals and is a luxury we are afforded in this
monograph. Such uniquely detailed code displays are intended to increase the
reader’s understandingof our usage of Mathematica(cid:2)R, assist in verifications, and
facilitatefurtherexperimentation.
Itisalsoworthmentioningherethatuponinitiallyimplementingthemethodof
Chap.3, it was also evident that the preliminary results were void of the level of
elegancethatmathematiciansstrivetoachieve.Therefore,anapproachwassought
that wouldsimultaneouslygive insights into existence/nonexistenceof the B-Type
1 orthogonalpolynomialsequences and also yield a tractable problemthat would
ultimatelyadmitelegantresults.Thesegoalswereaccomplishedusingsimplifying
assumptionsthatreducedtheproblemtoamanageableformatthatwasassimilarin
structureaspossibletotheB-Type0classthatShefferanalyzed.
To enhance the novel results of this monograph, in Chap.1 we additionally
includeanoverviewoftheresearchthatmotivatedtheirestablishment.Webeginby
addressingSheffer’sderivationoftheB-Type0generatingfunctiondefinedabove,
as wellas the characterizationsof the B-Type0 orthogonalsets. Since, in 1934,J.
Meixnerinitiallystudied thisgeneratingfunctionand determinedwhichsets were
also orthogonalusing a differentapproach than Sheffer,we also coverthe central
detailsofMeixner’smethodandresults.WethenbrieflyalludetoW.A.Al–Salams’s
extensionofMeixner’sanalysis.
Preface ix
Additionally,wediscussthattherewereactuallythreeclassificationsthatSheffer
developed:A-Type, B-Type (our focus in Chap.3), and C-Type. The discrepancies
betweenthese typeswill also be addressed.We thenpresenta summaryof the σ-
TypeclassificationdevelopedbyE.D.Rainville,whichisanextensionoftheSheffer
A-Typeclassification.Altogether,theShefferSequences,thenotionofType,andthe
relevantbackgroundmaterialareelucidatedinordertofacilitatethetransitioninto
thelattermaterial.
To enhance this monograph even further, in Chap.2 we discuss several of the
manyapplicationsthatclassicalorthogonalpolynomialssatisfy,whichincludefirst-
and second-order differential equations quantum mechanics, difference equations
and numerical integration. We first develop each of our applications in a general
contextandthenshowthespecificrolesplayedparticularA-Type0orthogonalsets.
Through covering each of these applications, we also develop additional funda-
mental terminologies, definitions, lemmas, theorems, etc. that are very important
in thefield oforthogonalpolynomialsandspecialfunctionsin a broadcontext.In
essence, Chaps.1 and 2 are intended to be used as (1) a concise, but informative,
reference for developing new results related to the A-Type 0 orthogonal sets and
classicalorthogonalpolynomialsin generaland (2)providematerialforadvanced
undergraduatecourses,orgraduatecourses,inpureandappliedanalysis.
For the benefit of the reader, each chapter is self-contained. In addition, with
respectto space constraints,this entiremonographhas beenwritten with as much
detail,rigor,andsupplementationviainformativeconcreteexamplesasfeasible.
This work represents the culmination of approximatelythree years of research
ontheShefferSequencesandrelatedstructures,whichwasconductedatPennState
Erie,TheBehrendCollege.
Lastly, the author would like to thank Blair R. Tuttle for assisting in the
proofreadingoftheoverviewoftheSchro¨dingerequationinChapter2andJennifer
K.UlrichforassistingwiththeadditionalproofreadingoftheBrief.
Erie,PA,USA DanielJosephGaliffa