PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 1, January 1996

AN GENERALIZATION OF THE q- THEOREM AND AN APPLICATION

YUNKANG LIU

(Communicated by Hal L. Smith)

Abstract. An integral identity which generalizes the q-binomial theorem is presented. This identity is used to determine the exponential expansion of the solution of an integro-differential equation.

1. Introduction One of the most important formulas for basic hypergeometric is given by the q-binomial theorem

∞ (a; q)n n (az; q) (1) φ (a; ; q, z)= z = ∞ , z < 1, q < 1, 1 0 (q; q) (z; q) − n=0 n | | | | X ∞ which was derived by Cauchy (1843), Heine (1847) and by other mathematicians (see, e.g., Gasper and Rahman [1]). Motivated by our investigation of the so-called pantograph integro-differential equation [3], we formulate the following integral identity n i 1 1 1 1 a − qj ∞ i=1 − j=1 1+ dµ(q1) dµ(qn) ··· n  i  ··· n=1 1 1 Q 1 Q qj (2) XZ− Z− i=1 − j=1 1 n   ∞ 1 a 1 q dµQ(q) Q = − − , 1 n n=0 1 R 1 q dµ(q) Y − − where a is a complex numberR and µ(q) a complex function of bounded variation over [ 1, 1]. For simplicity, all considered in this paper are of Riemann-Stieltjes type.− To guarantee convergence of the series and products in (2) we assume that µ satisfies 1 (3) dµ(q) < 1 1 | | Z− and 1 1 (4) dµ(q) < . 1 1 q | | ∞ Z− −| |

Received by the editors July 25, 1994. 1991 Mathematics Subject Classification. Primary 33D99; Secondary 45J05.

c 1996 American Mathematical Society

165

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 166 YUNKANG LIU

It is easy to see that (2) implies (1) by letting µ(q)=0forq [ 1,q0]andµ(q)=z ∈ − for q [q0, 1], where z < 1, q0 < 1. Denote∈ the left-hand| | side| of| (2) by Φ(a; µ( )). One consequence of (2) is the formula Φ(a; µ( ))Φ(b; aµ( )) = Φ(ab; µ·( )), where b is a complex constant. 1 · · · If we replace µ by a− µ in (2) and then let a we obtain the integral identity − →∞ 1 1 n n i ∞ i=1 qi − 1+ dµ(q1) dµ(qn) n i 1 ··· 1 ··· n=1 Z− Z− i=1Q1 j=1 qj (5) X − 1 ∞ Q  Q  = 1+ qndµ(q) , n=0 1 Y  Z−  where µ is a complex function of bounded variation over [ 1, 1] that obeys (4). − 2. The proof of (2) q n n Let µn(q)= 1p dµ(p), n 0, be such that dµn(q)=q dµ(q). It is straight- forward to verify− formally for any≥ constant n 0theidentity R ≥ (1 µn(1))Φ(a; µn( )) = (1 aµn(1))Φ(a; µn+1( )). − · − · From the preceding identity we derive by induction that N N (1 µn(1)) Φ(a; µ( )) = (1 aµn(1)) Φ(a; µN+1( )) "n=0 − # · "n=0 − # · Y Y holds for any N 0. Noting that limN Φ(a; µN+1( )) = 1, we obtain (2) by letting N in the≥ preceding identity. The→∞ preceding argument· can be made rigorous by exploiting→∞ our assumptions (3) and (4). For instance, the series on the left-hand side of (2) converges absolutely if (6) 1/n n i 1 1 1 1+ a − qj i=1 | | j=1 | | lim sup dµ(q1) dµ(qn) < 1. n  ··· n  i | |···| | 1 1 Q 1 Q qj →∞ Z− Z− i=1 − j=1 | |  To see that (3) and (4) implyQ (6), weQ choose a constant p (0, 1) such that  ∈  1 p 1 1+ a − 1+ a | | dµ(q) + | | dµ(q) dµ(q) . p 1 q | | 1 1+q | |≤ 1| | Z − Z− Z− By separating the integral domain [ 1, 1]n into [ p, p]n and [ 1, 1]n [ p, p]n,we obtain − − − − − n i 1 1 1 1+ a − qj i=1 | | j=1 | | dµ(q1) dµ(qn) ··· n  i | |···| | 1 1 Q 1 Q qj Z− Z− i=1 − j=1 | | p  n 1 p n ( a ; p)n Q Q 1+ a − 1+ a −| | dµ(q) + | | dµ(q) + | | dµ(q) ≤ (p; p)n p | | p 1 q | | 1 1+q | | Z−  Z n − Z−  ( a ; p) 1 −| | ∞ +1 dµ(q) , ≤ (p; p) 1 | |  ∞ Z−  which implies (6).

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INTEGRAL GENERALIZATION OF THE q-BINOMIAL THEOREM 167

3. An application of (2) and (5) Consider the pantograph integro-differential equation (7) 1 1 y0(t)=ay(t)+b y(qt)dµ(q)+c y0(qt)dµ(q),t>0,y(0) = 1, Z0 Z0 where a, b and c are complex constants, a =0,andµis a complex function of bounded variation over [0, 1] that obeys 6 1 n 1. c 0 q dµ(q) = 1, for all n =0,1,...; 1 6 1 1 2. b 0 dµ(q) < a , 0 1 q dµ(q) < ; | R| | | | | − | | ∞ 3. 1 qn dµ(q) 0asn . 0 R | |→ R →∞ UnderR the last assumption it is proved in [3] that the first one is both necessary and sufficient for the existence and uniqueness of a smooth solution of (7). In order to study the asymptotic behaviour of the solution of (7), we seek a solution of the form (8)

1 at ∞ n y(t)=α e + ( a)−  n=1 − 0 ···  X Z n i 1 1  i=1 b + ac j−=1 qj q1 qnat e ··· dµ(q1) dµ(qn) , n  i  ···  0 Q 1 Q qj Z i=1 − j=1    where α is a constant to beQ determined.Q It is trivial to verify that the preceding series satisfies (7) if and only if α obeys

n i 1 1 1 b + ac − qj ∞ n i=1 j=1 α 1+ ( a)− dµ(q1) dµ(qn) =1,  − ··· n  i  ···  n=1 0 0 Q 1 Q qj  X Z Z i=1 − j=1  which, by (2) if b =0orby(5)ifbQ= 0, impliesQ that   6  1 1 n ∞ 1+a b q dµ(q) (9) α = − 0 . 1 n n=0 1 c 0Rq dµ(q) Y − One obvious consequence of (8) and (9) isR that when a>0thesolutiony(t)of (7) increases (in modulus) exponentially. More specifically,<

1 1 n at ∞ 1+a− b 0 q dµ(q) lim y(t)e− = =0, t 1 n 6 →∞ n=0 1 c 0Rq dµ(q) Y − which partly proved a conjecture posed in [3].R Another consequence is that when a =0thesolutiony(t) of (7) is uniformly bounded on [0, ). We can also use (8) to< discuss the asymptotic behaviour of the solution in the case∞ of a<0. However, this case has already been studied in [3] under a slightly weaker< assumption and for the more general equation (10) 1 1 y0(t)=ay(t)+ y(qt)dµ1(q)+ y0(qt)dµ2(q),t>0,y(0) = 1, Z0 Z0

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 168 YUNKANG LIU

where µ1 and µ2 are complex functions of bounded variation over [0, 1]. Remark. An exponential expansion that is much more complicated than (8) does exist for the solution of (10). However, the corresponding integral identity is less elegant and the proof is too lengthy to be presented in this article. Acknowledgments The author wishes to thank his supervisor Dr. A. Iserles for many stimulating discussions on this subject. The author is supported by scholarships from Cam- bridge Overseas Trust, CVCP, Fitzwilliam College and C.T. Taylor Fund. References

1. G. Gasper and M. Rahman, Basic hypergeometric series, Cambridge Univ. Press, Cambridge, 1990. MR 91d:33034 2. A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math. 4 (1993), 1–38. MR 94f:34127 3. A. Iserles and Y. Liu, On pantograph integro-differential equations, J. Integral Equations Appl. 6 (1994), 213–237. MR 95g:45008

Department of Applied Mathematics and Theoretical Physics, University of Cam- bridge, Silver Street, Cambridge, CB3 9EW, England E-mail address: [email protected]

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