A NEW CLASS OF INFINITE PRODUCTS , AND EULER ’ S TOTIENT

We introduce some new infinite products, the simplest being(1−y)∏k=2∞∏j∈ϕk(1−ykqj)1/k=(1−y1−qy)1/(1−q),where ϕk is the set of positive integers less than and relatively prime to k, valid for |y|∧|qy| both less than unity, with q≠1. The idea of a q-analogue for the Euler totient function is suggested.

(1 --y) H H (1--ytCq')l/' _-- where Ck is the set of positive integers less than and relatively prime to k, with Yl and both less than unity, and q 1.This ties in with a far-reaching idea in analysis and combinatorics, the notion of a q-analogue.This notion is to replace, where appropriate, the integer parameter n by (1-q")/(1-q), n! by H= 1(1-q")/(1-q), etc..This is most familiar in the study of vector spaces over finite fields and the theory of hypergeometric series.Here it emerges in connection with a class of infinite products with fractional (and in general, arbitrarily small) exponents.
(1.2) n=l If x is replaced by y' and the product operator II is applied to both sides, we obtain II (i yk)(k)/k -/(-) (1.3) k=l by rearranging the product on the left and applying the identity where () .() for Euler's totient (n)= 16, I.If we now take logarithms in (1.3) and expand, we discover that (1.3) is the generating function expressing of the familiar identity the Mhbius inversion of (1.4).Now a q-analogue of (1.5) is q"q,(k In fact, the generating function expression of (1.6) that reduces to (1.3) for q 1 is (1.1).
It is not yet clear what properties (e.g., the multiplicativity) of have analogues for with q # 1.As a curiosity, we note that differentiation of (1.6) followed by q--l yields a Mhbius inversion of the shallow identity j he(n).A somewhat more impressive looking result han (1.1) is ((1 qy)( y)/O (1 T y) H H (1 + ykq,)l/k y)q(1 + qy)] (1.8) k 2 j valid for the same conditions as in (1.1).However, if each side of (1.8) is multiplied by the corresponding side of (1.1), what emerges is in fact a case of (1.1).
In the spirit of Euler we shall explore a variety of formal identities having the "style" of (1.1) while keeping details about domains of convergence and branching in the complex domain at a minimum.Our main result (of which (1.1) is a special case) is the Theorem of 2.
Reference [9] also examines the behavior of certain infinite products with fractional exponents as functions of a complex variable.[5] derives Euler products from results herein.where 4b is the set of positive integers less than and relatively prime to k.
PROOF.If we associate the sequence (c)= ezp(b,q) with (b,) so that the left side of (2.1) is expanded thus a + a2(1 + c/) + a3(1 + c/3 + el I) + (1 + / + d/ + c/ + cl/) + as(1 + c/6 + c/6 + c/ + c/s + c /6) we see that under certain conditions the terms rearrange into j=l j=l =1 j=l where the indices of the c terms are the countable set of Farey fractions in the interval (0,1).Since (2.5) is the same as the right side of (2.1) this completes the proof formally.The conditions which validate equivalence of (2.4) and (2.5) are trivially the criterion of the theorem that all the series of the theorem converge after (a) and (b) are chosen.End of proof.
(1.1) and (1.8) are representative of a large class of new infinite products which may be of interest in at least the Theory of Partitions.To abbreviate these products we define the product operator l-lj II II. (3.1) The following identities are simply derived from the theorem or by combining (1.1) and (1.2) variously with q, y, chosen so that all denominators are defined.
(l+y (l,+!#t:q'' 'It' ({1 ql/)l--+-( --1 --+-!/) '/('-ql, ( i--2-,/kIIj (l+y)kl'lj(l_(_y)}q,),/} (iI +Y I/('-i) +q!l] There would seem to be many interesting questions arising from these infinite products, such as the behavior of the branching effects of the fractional indices on the left sides of the identities, or the combinatorial interpretations.In exactly which sense the infinite products are true as functions of the complex variables seems worthy of further investigation.We remark that their numerical validity for various real q and /may be easily checked with a small calculator.
For q and y as previously, we evaluate )j'/k'" + k I-I j (1 yq where m is a positive integer.(1.1) relates to m =0 of (4.1).The following lemma is the case (a) y/k , (b) k (log q)/q of the main theorem.
The results of this paper ce only from considering ces of the Threm in 2 where (b) k(log q)/q.However, other choices of (b) may be of interest.In pticul, the author h found Jacobi theta function trsformations from the ces (b) k(log q)/q, d results for the ithmetic functions they generate.Numerous other products exist related to those stated herein.