0, 2

Internat. J. Math. & Math. Sci.

VOL. Ii NO. 4

(1988)

763-768 763

A PAIR OF BIORTHOGONAL POLYNOMIALS FOR THE

SZEGO-HERMITE WEIGHT FUNCTION

N.K. THAKARE

Mathematics Department University of Poona Pune 411 007, INDIA

and

M.C. MADHEKAR

Millnd College of Science Aurangabad 431 004, INDIA

(Received April I, 1987 and in revised form August 28, 1987)

ABSTRACT. A pair of polynomial sequences

{S(x’k)}

and

{T(x;k)}

^where

S(x;k)

is

n m n

of degree n in xk

and

T(x;k)

is of degree m in x, is constructed. It is shown m

that this pair is biorthogonal with respect to the Szeg-Hermite weight function

Ixl2exp(-x2),

( >-I/2) over the interval (--,-) in the sense that f

Ixl

^{2 exp(-x}

²⁾ S(x;k) T(x;k)dx

0 if m

#

n

0, tfm--n

where m,n 0,1,2 and k is an odd positive integer.

Generating functions, mixed recurrence relations for both these sets are obtained. For k I, both the above sets get reduced to the orthogonal polynomials introduced by professor Szeg.

KEYS WORDS AND PHRASES. Szego-Hermite weight function, Biorthogonal pair, Generating functions, Recurrence relations, etc.

1980 AMS SUBJECT CLASSIFICATION CODE. 33A65, 33A99, 42C05, 42C99.

I. INTRODUCTION.

The biorthogonality conditions are useful in the computations involving the penetration of gamma rays through matter as well as in determining the moments of a hypergeometric distribution function. The notion of biorthogonality dates back to Didon [I] and Deruyts [2]. The questions of constructing biorthogonal pairs of polynomials corresponding to the weight functions of classical orthogonal polynomials were taken up by Konhauser [3] for the Laguerre weight function x e--X by Toscano [4], Chai [5], Carlitz [6] and Madhekar and Thakare [7] for the Jacobi weight function (l-x) (l+x)8

and by Thakare and Madhekar [4] for the Hermite weight function

exp(-x2).

The SzegS-Hermite polynomials

H(x)

_n are orthogonal ^w.r.t, the Szeg-Hermite weight function

Ixl2exp(-x2),(

^> -1/2) over the interval (-,) and these are found

useful in connection with Gauss-Jacobi mechanical quadrature, see Szeg8 [8]. For O, Szeg-Hermite polynomials are just the classical Hermite polynomials.

2. A BIORTHOGONAL SYSTEM.

We shall construct a pair of biorthogonal polynomials w.r.t, the Szego-Hermite weight function

Ixl2exp(-x2),

> -1/2. Consider the following pair of polynomial sequences.

S(x;k) 2nF((kn +

k- ke)/2

+ +

e) n

(-l)J

^[n/2]

]j ^xnk-2kj/F

((kn+l+e)/2-kj+)" (2.1)

TV(x;

k)

(_i)[n/212

ⁿ

[n2] xn_2r/([n/2]_r)!

n r=O

(_l)S

^[n/2]-r

s=0 ^s

((2s+(k+l)e

+ 2v+l)/2k)[n/2

], ^(2.2)

where the value of is 0 or according to even or odd nature of n. Throughout this paper e always has this meaning; and [p] is the greatest integer less than or equal to p.

It is fairly easy to verify after reverting the order of summation for even and odd integers that

n n

x2kj

Sn(X;k ^(-l)n22n

^{r(kn++k/2) Z (-I)j} /r(kj++I/2)

j=0

J

(_l)n22n

n! [F(kn+v+k/2)/F(kn++I/2)] Zv-I/2

(x2;k);

(2.3) n

n _{n x}2kj+k

(x’k) (-I)ⁿ

22n+l

r(kn++l+k/2) E (-I)^j

S2n+l

j=O ^j

r

(kj++l+k/2)

(_l)n22n+In!

xk

zV+k/2 (x2;k);

_n (2.4)

n

22r

^r

T2n

^(x;k)

^(-l)n22n

^E

^--r!

^Z

^(_l)S rs ((s++I/2)/k)n’

r=0 s=0

(-l)n22n

n!

YV-I/2(x2;k),

n (2.5)

n r

T2n+ Iv

^(x;k)

^(-l)n22n+l

^l

(x2r+I/r’).

^I

(_l)S rs ((s+v+l+k/2)/k)n’

r=0 s=O

(-l)n22n+l

n! x

^yV+k/2

(x²;k). (2.6)

n

Here

Za(x;k)n

^and

Yna(X;k)

is a pair of Konhauser [3] biorthogonal polynomials w.r.t.

the Laguerre weight function

xcaexp(-x)

over (0,,) and are given by

n x

Zca(x’k)

^r(kn

⁺

^ca

⁺

^{I) n kj}

n n’ Z (-I)^j (2 7)

j--O j r(kj

+ +

I)

BIORTHOGONAL POLYNOMIALS FOR THE

SZEG6-HERMITE

WEIGHT FUNCTION 765

n r r

x s r

((s++l)/kn)"

see Carlitz [9]

Ye(x;k)

^{n r=Ol}

.

^{s=O7. (-i) s (2.8)}

where > -I, and k is a postive integer, and

a -x

ya

^F(kn+e+l) 6(n,m) with 6(n,m) (2 9)

I

x e Z (x;k) (x;k) dx

0 ^{n m} n!

the Kronecker’s delta. Using [I0] one readily obtains the following biorthogonality condition for the sets {S

(x;k)}

^{and {T}

(x;k)}:

n m

I Ixl

^{2u exp (-x}

2) S(x;k) TU(xk)

dx

n m

22n

[n/2]!

r(+e+(kn+k-ke)/2)

6(n,m) (2.10)

An independent proof of (2.10) is also possible by using the identity of Carlitz [9,

p. 249] m m-r

(-J)

m _r=O^7.

kj+c+m-rm_r

_s=O^{7. (-1 s}

m-rs

^{(s+c+ )/k)}

^m"

One has to note, however, that k is involved in

S(x;k)

and

T(x;k)

must be an odd

n m

positive integer in view of the existence theorem for blorthogonallty due to Konhauser [I0, p.255].

One readily obtains

+(k+l)/2

(x;k) and (2 II) r(kn+k++I/2) S (x;k) 2xk

F(kn++l+k/2)

S2n

2n+l

T (x;k) 2x _+(k+l)/2 (x;k) (2.12)

Zn+l

-I-2n

D S^B (x’k) 4 nk

xk-I F(kn+B+k/2)

B+(k-l)/2

(x;k) (2.13)

2n F(kn+B+ / 2) 2n-

3. SOME PROPERTIES.

Using the relationship (2.3) to (2.6) it is fairly easy to obtain many results for the Szeg-Hermite biorthogonal pair of polynomials from the known results for the Konhauser biorthogonal sets. The results stated below could also be proved directly.

Recall the Calvez and Ge’nin [II] generating function in the form (see also Srivastava 12

Z

m+nn ^Ym+n

^{(x;k) tn R}

(l++mk)exp{x(l-R) Y(xR;k)

^(3.1)

n=0

where m is any integer 0 and R (l-t)

-I/k.

By handling even and odd cases separately, from (2.5) and (2.6) respectively, one obtains

l

T2m+nV (x;k)tn/[n/2],.

^(3.2)

n=O

V (xU;k)] where

U=(l+4t2)

^-I/2k and

VU(+mk+(l+k)/2) [u-k

T

m(XU;k) ⁺

^t

T2m+l

V

exp{x2[l-(l+4t2)-I/k]}.

^The special case with m=O is worth noting. Using (3.2) for even case and then applying (2.12) one obtains in a combined form the recurrence relation for the second set

[n12]

)m

^2m

T(x-k) :

(-1 2

m0

nl2m ^(U-l)

m

Tn-2m(x;k)’ ^l#

^and ,>-I12.(3.3) Taking O, and n even in (3.3) and using the blorthogonality condition (2.10) we have the integral

I {x{

^{21 exp (-x}

²⁾ Sl2m

^(x;k)

^T2n(X;k)

^{dx (3.4)}

F(km+l+k/2) where with 0,

T2n(X;k)

^{is the}

(_i)ⁿ 4

m+n

(-n)m(-I/k)n_m

second biorthogonal set suggested by the Hermite polynomials; see Thakare and Madhekar [4]. The integral (3.4) says that

T2n(X;k)

^{are othogonal to}

Ixl21S 12m(x;k)

w.r.t, the weight function exp(-x

2)

when n ^> m+I/k.

Consider the generating function first given by Genin and Calvez [13]; (see also Karande and Thakare [14], Prabhakar [15]):

Z

(c)nZ

^(x;k)

tn/(l+a)kn

^{(l-t)-c F}k

txk/(l-t)k

^k (3.5)

n=O (k, l+a)

where

Itl

^{< and A(m,6) stands for} the sequence of parameters 6/m, (6+l)/m (6+m-l)/m, (m>l). Using (2.3) one obtains from (3.5), an expression involving

Sn(X;k)_

^which after putting to use relation (2.11) gives a corresponding even

relation for odd

S2n+l

^{(x;k) This} resulting expression further with the help of the relation

(x;k)

t2n+I/n!

^{(3 6)}

Z

(C)n s2n+

^(+k/2)nk

n=0

t(k+2+ke)/(k+2B) Z

(C)nSn+l(X;k)= t2t/n!(+l+k/2)nk

^{where8=t, d/dr}

n=0 yields

(c)n

n=O n!

(+k/2)nk

^{S (x;k)t}

2n+l 2tx^k

u-2k(l+c) (u-2k

^8ckt2

2n+l -)k+2 (3.7)

+

¹⁶

ckt3x3ku

^2k(c+2)

^+I;

^W

IFk (k,u+l+k/2);

(k+2) (l++k/2)

k

IFk

A(k,l++3kl2;

where W

4x2kt2/(l+4t2)k

^k

In fact, one obtains after combining even case with (3.7) the following generating function for the first blorthogonal set

{S(x;k)}:

n

(C)[n/2]

tn (+k/2)

u2kc V

^c;

W7

Z S (x;k)

(+I/2)

IFk

n=0 In/2]!

(+k/2)k[n/2]

ⁿ

(k,+i/2);

k+2u (k,

1++k12);

16 ckt3 x3k

U2k(c+2)

V

^c+l;

+

(k+3) (l++k/2)

k

iFk A(k,l++3k/2;

W

BIORTHOGONAL POLYNOMIALS FOR

THE SZEG-HERMITE

WEIGHT FUNCTION 767

We finally state the differential equation satisfied by the first set

{S(x;k)}

in the form

[x2(xD+2v+l+e)]k {xl-2k

(D-ek/x)

Sn(X;k)}

^(3.9)

(2x2)

^k {x D

S(x;k)

^nk

S(x;k)}

anda differential recurrence relation

n n

for the second set

k T (x;k) -2xD

T(x;k)

2(l+ml+2-2x

2) T(x;k)

(3.10)

n+2 n n

ACKNOWLEDGEMENTS. The authors are grateful to the referee for fruitful suggestions.

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