616 W. A. AL-SALAM [June

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1 616 W. A. AL-SALAM [June 2. Colin Clark C. A. Swanson, Comparison theorems for elliptic differential equations, Proc. Amer. Math. Soc. 16 (1965), F. R. Gantmacher, The theory of matrices, Vol. I, Chelsea, New York, Philip Hartman Aurel Wintner, On a comparison theorem for self-adjoint partial differential equations of elliptic type, Proc. Amer. Math. Soc. 6 (1955), Walter Leigh ton, Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), M. H. Protter, A comparison theorem for elliptic equations, Proc. Amer. Math. Soc. 10 (1959), The University of British Columbia g-analogues OF CAUCHY'S FORMULAS WALEED A. AL-SALAM 1. Let g be a given number let a be real or complex. The ath "basic number" is defined by means of [a] = (1 ga)/(l g). This has served as a basis for an extensive amount of literature in mathematics under such titles as Heine, basic, or g-series functions. The basic numbers also occur naturally in many theta identities. The works of Jackson (for bibliography see [2]) Hahn [3] have stimulated much interest in this field. One important operation that is intimately connected with basic series as well as with difference other functional equations is the g-derivative of a function /. This is defined by (i.i) /"(gx) f{x) mi*)- ; \\ x{q - 1) Jackson defined the operations, which he called g-integration, (1.2) f Xfit)d{q, = x(l -q)zz qkf{xqk) J 0 4=0 (1.3) f "fiodiq, t) = x(l -q)jz q~kfixq-k) J X 4=0 Received by the editors October 28, 1965, in revised form, November 30, 1965.

2 1966] ^-ANALOGUES OF CAUCHY'S FORMULAS 617 provided the series on the right h side are convergent. Both (1.2) (1.3) are inverse operations to (1.1) are analogues of the definite integrals fxf(t)dt fxf(t)dt respectively. In fact each approach the respective integral as q >l when / is Riemann integrable in the intervals of integration. The definite g-integral Jxaf(t)d(q, t) is defined by means of (1.4) f Xf(t)d(q, t)= f Xf(t)d(q, - f "f(t)d(q, /). The purpose of this note is to obtain g-analoques of Cauchy's formulas for multiple integrals (1.5) " /» X /» xn-x /» XI I j I /(/) dtdxi dxn-i i rx =- I (x - /)"-'/(/) dt (n-l)\ja JK> I - * _ /OO /» 00 /* 00 /(/) dtdxi dxn-i 1 r > =- (/ - x)»-y(/) dt. (n - l)\jx JKJ The g-analogues of (1.4) (1.5) are given in Theorems 1 2 below. We remark that (3.1) (3.2) can be regarded as a transformations of w-fold infinite series to a single infinite series. These are basically different from a finite analogue of (1.5) that has been recently given by Traub [4]. We further note that (3.1) (3.2) can be used to define fractional g-integral g-derivative in the same way that (1.4) (1.5) have been used to define fractional integrals derivative. This we shall give elsewhere. 2. Let for a nonnegative in teger n, [0]! = 1, [«]!= [n] [n l] [!] if n>0, let further, 0<g<l (a + b)0 = 1, (a + b)n = (a + b)(a + bq) (a + bq"'1) if n > 0. We shall also use the notation (a)0 = 1, (a)n = (1 - a)(l - aq) (1 - aqn~l) if n > 0,

3 618 W. A. AL-SALAM [June "»1 _ iq)n _ k A We first give the following lemmas: iq)kiq)n~k Lemma 1. For integers «S;1 m^o we have m (nm+1) (2.1) Am = zz qjw - qm+l)n-i = f - y-o 1 - qn Proof. We have oo oo oo zz Amum = zzumzz q'iq3' - gm+1)-i so we have 00 oo = zz usqnj zz (qm+1)n-ium j=0 OT=0 1 v^ (g)m+n-l =- Z-j-«"", 1 uqn m=0 (q)m (2.2) f^-j^lf**^. m-o 1 uqn m=0 (g) Let us recall the formula [l, p. 66] V (a)k 4 TT 1 ~ au1t *- 7T u = 11-> k-0 (q)k r=0 1 Uqr so that (2.2) can now be written as - iq)n-i " 1 - «g'+" e 4m««= - - n - m-o 1 uq" r=o 1 uqr " 1 - «g»+h-l ~ (g"+1)4 = (q)n-i 11 -= iq)n-i.. «*. r=o 1 uqt k=o (q)k Equating coefficients of um we get (2.1). Lemma 2. For integral m 2; 1 we have (2.3) r'cr"-1 - rj)n-i = -^-(?-%. y-i 1 - gn The proof of this lemma is similar to that of Lemma 1.

4 1966] ^-ANALOGUES OF CAUCHY'S FORMULAS 619 Lemma 3. For nonnegative integer n, OO an+l(l - g") Y?'"(?' - qk+l)n-i i-o (2.4) ax(l - g") Y ik*0' ~ aqk+l) i an+1(qk+1)n i-l = a(x aqk+1)n. Proof. To prove this lemma we recall the identity due to Euler (2.5) (aa-b)n=y (-1)' q^-»i2a"-*b', r-0 L r J so that " :zj r» - n gr(4+1) E qj(qj ~ qk+%-i = Y (- i)r gr(r"1>/2 J- y-o r=o L r J 1 - g"~r Hence we get OO an+1(i - qn) Y q3'(q3' - qk+l)n-i i-o (2.6) ;=} r«-] = an+1 Y (~iy qr(r-l)l2gr(k+l) r=0 L r J Similarly = a»+l(1 _ <H-l)n _ (_ J_)n^n(n-l)/2+n(W-l)- (2-7) oo ax(l - qn) Y a'(xq' aqk+l)n-i = a[(x - aqk+1)n - (-l)«9a/2)»(n-i)+r.(*+i)ffl»]_ Substituting (2.6) (2.7) in the left h side of (2.4) we get the right h side of (2.4). 3. We now prove our main results. Theorem (3-D 1. // n 2:1 is a given integer / X /» Zn-l /» XI /-/(*) = f(t)d(q, t)d(q, Xi) d(q, X,.,) ^ J' 1 rx = -r--t7 (x - qt)n-if(t)d(q, t), in 1! Ja

5 620 W. A. AL-SALAM [June Theorem 2. If n^l is an integer I /» OO f* CO /» CO x J x -l " X, I /(0<*(<Z, t)diq, Xi) diq, Xn-i) (3.2) V ' 0-(l/2)n(n-l> /.oo = ~r-7tt (< - x)n-ifitqi-»)diq, t). [n 1J1 J, Both of these theorems can be proved by induction. We give the proof of the first as the proof of the second is similar. To prove Theorem 1 assume (3.1) is true for n = N. We have IN+lI& = r,!"?!, (i ^ ~ kk+l)n~ifitqk) [N 1J! J a a{t - aqk+l)n-if{aqk)}d{q, t) = r"!"^2, i i* W1 ~ qk+lh-i{*"+yifi*<f+3') [N 1J 1 4=0 J=0 - an+1qn'fiaqk+i)] _ 7^-7T ^ E qk+'fiaqk){axixqi - aqk+1)n.i [N 1J fc=0 ;=0 - a»+v - g*+%_i}. This reduces after some simplification to 00 OO [N - 1]!/JV+1/(x) = (1 - q)2 zz q fixq )ixn+1 ~ an+1) IZ fitf ~ S^V-i 8=0 j^o OO - (1 - q)2 IZ q'fiaqs)ax zz?y(*?y- a?s+1k-i oo «=i «o + (1- q2) JZ q fiaq>)a^ zz «'(?'- <T+1V-i. «=o Evaluating the inside sums in the right h side of the above equation by means of Lemmas 1 3 we get (1 q) M IN+1fix) = ^ zz W*0 _ xq*l)nfixq>) - o(x - aqs+1)nfiaq<)} [N\!»=o = JnTiI (x-9t)nf(l)diq,t). i O

6 i966]?-anal0gues OF CAUCHY'S FORMULAS 621 Since (3.1) is true for w = l our proof is complete. We remark that Lemma 2 is required in the proof of Theorem 2. References 1. W. N. Bailey, Generalized hypergeometric series, Cambridge Tract No. 32, Cambridge, T. W. Chaundy, Frank Hilton Jackson (obituary), J. London Math. Soc. 37 (1962), W. Hahn, Uber die hoheren Heineschen Reihen und eine eintliche Theorie der sogennanten spezielen Funktionen, Math. Nachr. 3 (1950), J. F. Traub, Generalized sequences with applications to the discrete calculus, Math. Comp. 19 (1965), University of Alberta at Calgary