expansions of hypergeometric functions in hypergeometric functions

Expansions of Hypergeometric Functions inHypergeometric Functions

By Jerry L. Fields and Jet Wimp

Abstract. In [1] Luke gave an expansion of the confluent hypergeometric func-

tion in terms of the modified Bessel functions Iv(z). The existence of other, similar

expansions implied that more general expansions might exist. Such was the case.

Here multiplication type expansions of low-order hypergeometric functions in

terms of other hypergeometric functions are generalized by Laplace transform

techniques.

1. General Expansions. The generalized hypergeometric function PFq(z), [2],

is defined by

pII («y)* k

(1.1)

PFq(z)=PFjai'---'^\z) = f: t±

T(a + p)

frt'IKM*y=i

where (_■)„ =I» '

We assume that no a, is equal to any b,- and that no b¡ is a negative integer. For

ease in writing, we employ the contracted notation

Thus (ap)k is to be interpreted as JJjLi (ai)k and similarly for (bq)k • Considered

as a power series in z, PFq(z) has a radius of convergence equal to infinity if p ^ q

and equal to unity if p = q + 1. In general, PFq(z) is not defined if p ^ q + 2.

However, in this paper, we shall say that pFq(zw) is equal to another series if the

coefficients of (w)k on both sides are equal, regardless of the relationship between

p and q.

Our first expansion is

(ap, crI \ _ <A (aP)n(a)n(ß)n(-z)H^t q+s y^ ^ ̂ I zwj - £ ________

/n + a, n + ß, n + apI \ /-n, n + y, cr\ \X p+2f 9+1 1 o i 11 i J. \Z I r+2f s+2 I n J W I •

\ 2n + y + l, n 4- bq \ ) \ a,ß,ds \ J

From (13) we may derive other expansions by confluence:

„ (aP,cr\ \ \^ (ap)n(a)„(—z)nrt«+s I A j \zw ) = 2s 7TT-?-1-\-i

\bq,ds\ ) n=o (bq)n(y + ri)nn\

( n + a,n + ap \\ (-n, n + y, cr I \X p+li* 9+1 l o i ii i I. \Z I r-yïC s+1 I r \W I ,

\2n + 7 + 1, n + bq \ ) \ a,d, )

Received September 30, 1960; revised February 20, 1961. This work was sponsored by the

United States Navy through the Applied Mathematics Laboratory of the David W. Taylor

Model Basin.

p+

(1.4)

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

(1.5)

p+(1.6)

EXPANSIONS OF HYPERGEOMETRIC FUNCTIONS 391

(ap,cr \ y^ (ap)n(a)n(-z)n [n + a,n + ap\\

\09»«s / «=o (bq)nnl " "\ n + í>3 | /

p /""' Cr ,„A - V (ap)n(-z)n „ {n + apI \

rK+s \bq,ds ZW) 'h (bq)nn\ pK \n + bq\Z)

xr+iPs(~^'Cr|^.

For example, if in (1.3) we replace r by r + 1, 2 by z/cr+1, set ß = cr+, and

let ß —> œ, (1.4) results; (1.5) and (1.6) may be similarly derived; (1.6) is a result

given previously by Meijer, [7], 1953, page 355, but it is the only result above which

can be deduced from his work.

Equation (1.3) is proved by induction on p, q, r, and s. The case p = q = r =

s = 0 reduces to

(1.7) ezw - V (a)n(ß)n(-z)n (n + a, n + ßI \ -, f-n, n + y\ \

»tí (7 + n),»! \2n + 7 + 1 I / \ », ß /

a result given by Luke, see (1.8) of [3]. The proof of (1.3) then proceeds by Laplace

and inverse-Laplace transform techniques. Assume (1.3) true for p, q, r and s,

multiply both sides of (1.3) by (w)'~l, take the Laplace transform of both sides

with respect to w, and we obtain, see (17), page 219 of [4],

C -Xu, »-1 e, (ap,Cr \, r(er) (ap,Cr,tr\z\

Ie w ^F^{bq,dszw)dw = l^^F^{bq,ds \x)

fi»ï _ r(<r) y (ap)n(a)n(ß)n(-z)n (n + a, n + ß, n + ap\ \

U-8J ' (X)' h (bq)n(y + n)nn\ p+2t q+1 \ 2n + y + 1, n + bq \Z)

(-n,n + y,cr,o-\l\X 7+3? s+2 l o j 7- I .

\ a, ß, as IX/

The induction on r is completed by replacing 1/X by w in (1.10). The induction

with respect to s is effected by multiplying both sides of (1.3) by (w)°, letting

w = 1/X and applying the inverse-Laplace transform, see (1), page 297 of [4].

If the above Laplace transform techniques are applied to z instead of w, the in-

ductions on p and q can be similarly effected.

2. Specialized Expansions. In this section we give several interesting cases of

(1.3H1.6).From (1.3) we have

Jc,| \ 2y f. (yU-z)n

[1 + (1 - zyi*]y n=o n![l + (1 - z)1'2]2»(2.1) . .

/ -n,n + y,cr \ \

X'+2F°+2\y/2,(y+l)/2,ds\W)>

where we have used the relation, see [2], page 101, (6),


(2.2)

The expansion

JERRY L. FIELDS AND JET WIMP

(2.3)

^'(X\m)mm^{^ilM^^$)

+2K-1)n (n + y) 1 \ n (—n.n + 2y.cr\ \T fz\

(n)27r+2p+,^ ,' + T)d; Iw)in+yy

follows from (1.4), where 7,(z) is the modified Bessel function of the first kind.

Here, use is made of the expression

(2.4) r(, +1) I,(x = (x/2Ye~x ,F, (J + ¿ I 2x).

Also from (1.4) we get

pFí\bq\ W)~P+1 F°+1\a + ß + 2,bq\Z)

(aP)n(z)n „ ( n + ß + l,n + ap \\

X P„<a,",(2w - 1),

(2.5)+ E

í (bq)n(n + a + ß + 1)„

where

(2.6)p i«.»^\ _ < iv d + I3)" p //_n> « + « + /3 + 1 1 + x\P» (x)-(-l) n[ 2F,^ 1 + j8 -^-j

The Pn(a,ß)(x) are known as the Jacobi polynomials, see [8], and reduce to the

Chebyshev polynomials of the first kind if a = ß = — \.

Equation (1.5) yields the following expansions:

«•" '-(¿ih) - „^-"-»'"(".rl")'(2.8) ^(jJ-)-(i-»)"|.^C-é1)"*".«(;X|«'); ««i,nn\ n ( cr \ S V* 7(5 + n,z) ri /~ W, Cr| \

where

(2.10)

7(0, z) = r(o) - / e Y l di,

\a + 11 /

i9in p iaJ>\,„\ V (°p)n(-2)n p /w + a + l»n + flp|J\ r <Y„,ï(2.11) ^.^J^»^-—^^,^ n + K \z}Ln(w),



where

(2.12) Lna(x) = __L__» lFl ( -« IXVn! \1 + a | /

The Lna(x) are known as the generalized Laguerre polynomials, see [8].

The result (2.7) can also be deduced from the work of Rain ville [5], page 267,

(25), and (2.8) is a generalization of a result given by Chaundy [6] and Meijer

[7], 1952, page 483.If we use the formula

(2.13) „Fi (v + 11^) = r(„ + 1) (?y /,(*),

(1.6) yields the expansion

(2.14)

= r( 1 + S) (?Y ± ^^ h+n(z) r+1Ps (~n' Cr I w)\z/ n=o nl \ a„ /

Since the merits of modified Bessel function expansions are well known, we

consider the convergence properties of (2.3) in more detail. They are determined

principally by the asymptotic properties of the polynomials

(-n,n + 2y,cr\ \

for large n. For an extensive treatment of the asymptotic properties of these poly-

nomials for large n, see [9]. In general, the convergence properties of (2.3) are

superior to the original series definition if the polynomials

/-n,n + 2y,cr\ \

r+*F°+1\ è + 7,ds \W)

are interpolatory or nearly so. For example, if r = s, w = 1 and c¿ — c,- is never

an integer or zero, for i j¿ j,

,+,,,+,(-»■ ;+%*|i) - [_■ +1 *,„-•].[, + o(i)].r r

P = zZ ct — Lu dt,(-1 í—1

and the A and P('s are constants independent of n. Incorporating the inequality,

see (1), page 49 of [10],

(2.16) | L(z) | ^ r}^ri} exp ilm(z)},

we see that (2.3) converges like

.A ((n}2»+y JU ny~2ct

<2-17) Sw+&


394 JERRY L. FIELDS AND JET WIMP

The original series definition, however, converges like

<2-18> 5 soo ¡i

' \z

3. Further Expansions Involving Free Parameters. If in (1.3), we replace

p by p — 2 and set a = ap_,, ß = ap , w = 0, we get

r q , \ i _ V (ap)n(-z)" v ( n + ap I \{i'1} ~ h (KUy + n)nn\ P"q+l \n + bq, 2n + y + 1 \z) '

Then replacing ap by ap + k, bq by bq + k, y by y + 2k and multiplying both sides

of (3.1) by

(Cr)k(zw)k

(ds)kk\ '

we get

(cr)k(zw)k _ (cr)k(zw)k J (fc + oP)„( -z)n

(ds)kk\ (ds)kkl n=o (fc + bq)n(n + 2k + y)nn

(3.2)

y p / n-|-fc + ap I \X ^3+1 Vn + fc + bq, 2n + 2k + 7 + 117

y> (Oj)n(—z)n p / «■ + ap I \

¿3 (Ö9)„(7 + n)nn! pr9+1 V« + K, 2n + y 4- 11 Z)

X(— n)k(n + y)k(bq)k(cr)k(w)k

(3.3)

(Oj.)*(d.)*Ü5!

Summing the terms represented in (3.2) from k equal zero to infinity, and inter-

changing the summation processes with respect to fc and n, we obtain

FAr| \_y (aP)n(-zY ( n + ap \\

rtt \ds |ZW) nh (bq)n(y + n)nn\ p**+l\n + bq,2n + y + l\Z)

/-n,n + y,bq,cr\ \

X'+'+!F'H ap,ds \W)-

Using the Laplace transform techniques of Section 1, we arrive at the expansion

F (c"e' I ~,„\ _ V (et)n(aP)n(-z)n

r+t*s+u \d., fu IZW) ¿Í (fu)n(bq)n(y +»)„»!

foA\ v p ( n + ap,n + et \\(3.4) X P+ttq+u+l yn + bq;n+fuj2n + y + 1\z)

(-n.n + y,bq,cr\ \

X3+rÄ,i ap,ds \W)-

Equation (3.4) and its confluent form not containing y are generalizations of results

given by Chaundy, see (11), page 187 of [2].

As a final example of how Laplace transform techniques may be used in general

expansions, we prove



AÍfW^Í_W-2í%¥^\°Q I / \««l / n=<> (bq)nn\

„ í-n, l-n-bq,cr (-l)^+1p\

Xr+3+1 P+a\ 1 -n-ap,ds X /•

Again the proof proceeds by induction on p, q, r and s. The case p = q = r=s = Q

reduces to

(3.5)

\z uze -e = _:

(X + pYz"

(3.6)

n-o n! \ X/

since

(3.7) iF0(a\z) = (1 -•)—.

Assuming (3.5) true for p, q, r and s, the inductions with respect to r and s can

be effected by using with respect to p the Laplace transform techniques illustrated

in the proof of (1.3). The inductions with respect to p and q are similarly carried

out with respect to X after making use of the relationship

an F (~n'aA\ (ap)n(-z)n jp (~n,l - n - bq (-l)q+p\

(3.8) ,+PF^ K \z)= {hq)n »JP*\ l-n-ap ^^z~)-

This completes the induction proof. Equation (3.5) is a generalization of results

given by Chaundy, see (12)-(15), page 187 [2]*.

Midwest Research Institute

Kansas City, Missouri

1. Y. L. Luke, "Expansion of the confluent hypergeometric function in series of Besselfunctions," MTAC, v. 13, 1959, p. 261-271.

2. A. Erdélyi, W. Magnus, F. Obebhettinger & F. G. Tricomi, Higher TranscendentalFunctions, McGraw-Hill Book Company, Inc., 1953, Vol. 1. Hereafter in this list, we use theabbreviation H.T.F.

3. Y. L. Luke & Richard L. Coleman, "Expansion of hypergeometric functions in seriesof other hypergeometric functions," Math. Comp., v. 15, 1961, p. 233.

4. A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Tables of Integral Trans-forms, Vol. 1, McGraw-Hill Book Company, Inc., 1954.

5. E. D. Rainville, "Certain generating functions and their associated polynomials,"Amer. Math. Monthly, v. 52, No. 5, May 1945, or H.T.F., Vol. 3, p. 239-250!

6. T. W. Chaundy, "An extension of hypergeometric functions," Quart. J. Math. OxfordSer. 14, 1943, p. 55-78.

7. C. S. Meijer, "Expansion theorems for the G-function," Indag. Math., v. 14-17,1952-55.8. H.T.F., Vol. 2.9. J. L. Fields & Y. L. Luke, "Asymptotic expansions of a class of hypergeometric poly-

nomials with respect to the order," Midwest Research Institute Technical Report, July 1,1959.10. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University

Press, 1945.

* There are two misprints in these formulas. In (12), p/g should be replaced by q/p and

in (13), -a' by a' on the right side of the equation.


expansions of hypergeometric functions in hypergeometric functions

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