Hypergeometric series recurrence relations and some new orthogonal functions

69 Pages·1978·1.907 MB·English

by  James Arthur Wilson

HHYYPPEERRGGEEOOMMEETTRRIICC SSEERRIIEESS RREECCUURRRREENNCCEE RREELLAATTIIOONNSS AANNDD SSOOMMEE NNEEWW OORRTTHHOOGGOONNAAL FUNCTIONS.
WILSON, JAMES ARTHURWILSON,JANIESARTHUR
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' WILSON, James Arthur, 1951-
HYPERGEOMETRIC SERIES RECURRENCE RELATIONS AND
SOME NEWORTHOGONAL FUNCTIONS
The Universityof Wisconsin—Madison, Ph.D., 1978
Mathematics
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UNIVERSITYMICROFILMS
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HYPERGEOME‘I‘RIC SERIES RECURRENCE RELATIONS
AND SOME NEW ORTHOGONALFUNCTIONS
A thesis submitted to the Graduate School of the
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JAMES ARTHURWILSON
Degree to be awarded: December 1.9 May 1.9 August 1915
Approved by ThesisReading Committee:
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HYPEMBOMETRIC SERIES RECURRENCERELATIONS
AND SOME NEW ORTHOGONALFUNCTIONS
A thesis submittedtotheGraduate School of the
Universityof Wisconsin—Madison inpartial fulfillment of
the requirements for the degree of Doctor of Philosophy
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______._JA_M_ES_A_R_TH—UR—WILS—ON——-—-—
Degree tobe awarded: Decanber 19 May 19 August 19.73..
Approved by ThesisReadingCommittee:
(Mm Clo/55% mm Is, Im
Major Professor Date df Examination
Dean,Jeraduate School
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HYPERGEOMETRIC SERIES
RECURRENCE RELATIONS
AND SOME NEW
ORTHDGONAL FUNCTIONS
by
JAMES ARTHURWILSON
A thesis submitted in partial fulfillment of the
requirements for the degree of
DOCTOROF PHILOSOPHY
(Mathematics)
at the
UNIVERSITY OF WISCONSIN-MADISON
1978
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TABLE OF CONTENTS
ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . . . . . . ii
I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . 1
II. 4173 POLYNOMIAL ORTHOGONALITIES. . . . . . . . . . . . . . . 14
III. GRAMDETERMINANTS . . . . . . . . . . . . . . . . . . . . . 31
IV. THREE—TERM CONTIGUOUS RELATIONS . . . . . . . . . . . . . . 43
BIBLIOGRAPHY..........................62
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ACKNOWLEDGMENTS
For creating the atmosphere inwhich all the ideas in this thesis
sprang up inevitably, my gratitude goes to Professor Richard Askey.
I also wish to thankProfessors Dennis Stanton andGeorgeAndrews for
their valuable contributions to this environment, my parents and my
wifeRosemary for their understanding and encouragement, and the fol-
lowing people for their inspiring examples: Burl Cannon and Professors
Raymond Redheffet, Kirby Baker, Basil Gordon, Richard Arena, Alfred
Hales, Theodore Motzkin, and Carl deBoor.
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I. INTRODUCTION
This thesis is largely the result of efforts to better understand
the connection betweenhypergeometric series and orthogonal polyno-
mials. Ahypergeomerric series is a series
al,...,aP; W (a1)k...(a )k k
pFq(b1,...,bq 2) - k—EO (b1)k...(bq)k(1)k 2 ,
with (a)k= a(a+1)...(a+k—1) if k11, and (a)o = 1. We call (a)ka
shifted factorial, since (1)k= M. It converges for all z if q1p;
and for ‘2] < 1 if q = p—l. Orthogonal polynomials forwhich explicit
symbolic Calculations canbe carried out due to the availability of
explicit formulas such as orthogonality relations, recurrence rela—
tions, and differential or difference equations, seem invariably to
involve hypergeometric series or q-series, generalizations of hyper—
geometric series. The polynomials can be expressed as hypergeometric
series or q—series, and their properties are consequences of hyper-
geometric series theorems or their q—extensions. This situation is
perhaps not so surprising, since a hypergeometric series is simply an
s
infinite series 2 tk with tk+1Itka rational functionof k, and
series more complicated than this are difficult to workwith. (Excep—
tions are the q—series, whichhave tk+litk a rational function of qk.
Many theorems for hypergeometric series generalize to q—series.)
However, what is more striking is that nearly all the special
types of hypergeometric series forwhich summation or transformation
formulas exist are involved in an essential waywith some orthogonal
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polynomials.
In the first chapter, we elaborate on these ideaswhile giving
necessary background information onhypergeometric series and ortho-
gonal polynomials. General references formost of theorthogonal
polynomial information are Erdélyi [5], volume 2, and Szegfi [11].
Hypergeometric series references are Bailey [1] and Erdélyi [5],
volume 1. In succeeding chapters, we introduce somenew families
of orthogonal polynomials and biorthogonal rational functions whose
basic properties involve the deeper series identities, including some
new three-series relations. Wewill not be concerned herewith q-
series. Extensions to q-series arebeingworked out for all thenew
results inthis thesis.
The orthogonal polynomials knownmost widely are the classical
polynomials named for Jacobi, Laguerre and Hermite, with the following
explicit representations and orthogonality relations:
Jacobi Eolmmials:
((2,8) _ (n+1)n -n,n+o+B+1; l—_x
Pn (x) _ n! 2F1(o+l 2) ’
(1,3 > —1, n10;
(1.1) f1 (l—x)a(1+x)BPn(u’B)(x) rm( “e)(1:) dx - o, m54 11 ;
-1
Laguerrenolflomials:
(o+l)rl (—n;
Lga><x) TF1 u+1")-‘"'1"‘i° ‘
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f tue—c LIE“)(x) Lg“)(x) dx = 0, m7‘ n;
0
HermiteEolmomials:
112nm) - (-1)“22“n: 1.5149(x2), anus) = (—1)“22n+lnafNXZ),
n10;
w _x2
I 2 Hub!) Rubi) dx = 0, msén.
Among the useful explicit formulas satisfied by the classical poly—
nomials are recurrencerelations, differentiation formulas, second
order linear differential equations, and Rodrigues—type formulas.
For eitample, the Jacobi polynomials satisfy:
(1.2) xPIEu’B)(x) = Anrflj‘riflNx) +BnPrEa’B)(x) +cnPffiBNx) ,
with
A = 2(n+1)(n+a+e+1)
n (2n+d+8+1)2
_ 52-03
Bn ' (2n+o.+S)(2n+a.+B+2) ’
and
c = 2(n+a>(n+s) ,
n (2n+a+6)2 ’
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(1.3) d (,5) n+u+8+1 (a+1,B+l) .
5911“ (x) a 2 Pn—l (x) ’
(1.4) (1-x2)Cdd—l2z'2‘“ ”(2‘) + [s-a-(oc+8+2)x] Ev§“'5)(x) +
+n(n+oc+5+1) 9(Q’B)(x) - 0;
and
(1.5) (1—x)“(1+x)BP§“’B’(x) = (3—1)“—{(1-x°‘)*“(1+x)“*“}.
2n11! flat“
There are other orthogonal polynomials whichmakevery nice dis—
crete analogs of the classical polynomials. They satisfy similar
explicit formulas inwhich the derivative operator i: replaced by
the ordinary difference operator Af(x) = f(x+1) - f(x). These poly-
nomials and their orthogonality relations are:
Hahnpolynomials:
_ _ -n,n+o,+8+1,-x-
Qn(x’°’s’m ' 3F2(u+1,-N ' 1) ’
a,S>-loro.,B<—N,0_<_nil\l;
N (a.+1)x(—N)x . _ .
(1.6) =20W Qn(xw,5,N) Qm(x,u,B,N) 0, m7‘ 11,
X.
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