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A product formula for Eulerian functions and a general summation method are given. The q-Eulerian function Hk(xql,. q.) may be defined symbolically by Hk = x iN u (1 + qjH) if k > 1; in addition, Ho1. Roselle [4, (3.2) and (3.9)] proved the following: C0 (1) x k (x | ql ** qk) q= q 1 k rnx(l.^>k : nl, .,nk=O (2) U (X1 q qk =(_1)k xq 1 * qkk) These formulas are useful in obtaining the following: 0o t E I (lmin(ni, .., n ))mq," q n zt 3)t=0 n 1I,?. nk=? Z1)1 (_ 1)k +mzfl (z I q 1, q2 I qk )H(q q ... k, Proof of (3) is as follows. The left side becomes 00 t min(nl,...,nk)m ZE q7 1 q ... q kzt t=0 nl,...,nk=-j=1 = (1_Z)1 Z qnl qnkz max(nI..--nk) n 1.*.,nk-1?<j<nlm,.. 2 ,k j=1n...nk? Ij/m ao~~~~ k (_z)I E E q ln qqnkzmax(n1 .. nk)(q lqz)( 1/rn) j=lI n I nkmaxnl.,.nk/M Received by the editors December 19, 1974. AMS (MOS) subject classifications (1970). Primary 10A40; Secondary 05A19.

A summation formula and some properties of Eulerian functions