WDS> So is what Gosper now wants, this?: PRODUCT(over n & k with n>0, 0<k<n, gcd(k,n)=1)OF (z-k/n) or is it this?: PRODUCT(over k with 0<k<n, gcd(k,n)=1)OF (z-k/n)! Anyway, about inclusion exclusion and other games with residues mod n: A. I recently rediscovered, which was known to Gauss, the facts that the residues mod n which are relatively prime to n, (1) form a multiplicative abelian group, (2) which is cyclic if and only if n is odd-prime-power, 2 times odd-prime-power, or n=2 or n=4, (3) which is "bicyclic" if n>4 is a power of 2, that is, all such residues are powers of 5 times +1 or -1, and the powers of 5 times +1 constitute a cyclic subgroup of half the full size, where full size of course is EulerTotient(n). These facts often allow interesting results to be got for these "nice" n. B. Assume n is squarefree for simplicity. Let "weight(x)" be the number of primes in x's factorization. Use the fact that SUM(0<=k<=n) Binomial(n,k) * (-1)^k = (1-1)^n = 0. Conclude SUM(divisors d of n) (-1)^weight(d) = 0. Also SUM(divisors d of n with d>1) (-1)^weight(d) = -1. The application of this is SUM(k with 0<k<n and gcd(k,n)=1)OF something = SUM(divisors d of n) (-1)^weight(d) * SUM(k with 0<k<n and d divides k) something C. Products are sums if you use logs. That is, B and C allow you to do inclusion-exclusion where everything involved is a sum over an additive subgroup of residues mod n, i.e. nice. The function (-1)^weight(d) more generally can be replaced by MobiusFunction(d) if we allow n that are not squarefree -- that will work for many uses. <wds --------------------- We never actually understood this, and had to rediscover it ourselves: gamid[n_Integer] := Product[Gamma[z + k/n], {k, Select[Range[n], CoprimeQ[#, n] &]}] == Product[(k^(1/2 - k z) (2 \[Pi])^(1/2 (-1 + k)) Gamma[k z])^ MoebiusMu[n/k], {k, Divisors[n]}] Test: In[38]:= gamid/@{4,6,8,9,10,12,15,16} Out[38]= {Gamma[1/4+z] Gamma[3/4+z]==(2^(3/2-6 z) \[Pi] Gamma[4 z])/Gamma[2 z],Gamma[1/6+z] Gamma[5/6+z]==(2^(1-4 z) 3^(-3 z) \[Pi] Gamma[z] Gamma[6 z])/(Gamma[2 z] Gamma[3 z]),Gamma[1/8+z] Gamma[3/8+z] Gamma[5/8+z] Gamma[7/8+z]==(2^(5/2-16 z) \[Pi]^2 Gamma[8 z])/Gamma[4 z],Gamma[1/9+z] Gamma[2/9+z] Gamma[4/9+z] Gamma[5/9+z] Gamma[7/9+z] Gamma[8/9+z]==(8 3^(1/2-15 z) \[Pi]^3 Gamma[9 z])/Gamma[3 z],Gamma[1/10+z] Gamma[3/10+z] Gamma[7/10+z] Gamma[9/10+z]==(2^(2-8 z) 5^(-5 z) \[Pi]^2 Gamma[z] Gamma[10 z])/(Gamma[2 z] Gamma[5 z]),Gamma[1/12+z] Gamma[5/12+z] Gamma[7/12+z] Gamma[11/12+z]==(2^(2-12 z) 3^(-6 z) \[Pi]^2 Gamma[2 z] Gamma[12 z])/(Gamma[4 z] Gamma[6 z]),Gamma[1/15+z] Gamma[2/15+z] Gamma[4/15+z] Gamma[7/15+z] Gamma[8/15+z] Gamma[11/15+z] Gamma[13/15+z] Gamma[14/15+z]==(16 3^(-12 z) 5^(-10 z) \[Pi]^4 Gamma[z] Gamma[15 z])/(Gamma[3 z] Gamma[5 z]),Gamma[1/16+z] Gamma[3/16+z] Gamma[5/16+z] Gamma[7/16+z] Gamma[9/16+z] Gamma[11/16+z] Gamma[13/16+z] Gamma[15/16+z]==(2^(9/2-40 z) \[Pi]^4 Gamma[16 z])/Gamma[8 z]} With respectful thanks nevertheless, --rwg wds> Gosper has an annoying habit of using non-ASCII characters I cannot read, which come out as "?" on my screen. Or maybe they really are "?". Who knows? hmm, that might explain why I'm not getting any answers. Off list I've been merrily exchanging emails with Chinese and all sorts of exotica from the Special Characters menu. But this time the culprit is not math-fun--I can see the Uppercase Gamma in my Subject summary window, where there's a ? in Warren's reply. Who else is having this problem? Was anybody besides Warren pwned by my BLACK RIGHT-POINTING SMALL TRIANGLE (?), (which GMail has suddenly stopped inserting on double-clicking the font table)? Maybe Warren is alone on a network made of juice cans and string? Anyway, he's right. I should've stipulated 0<k<n under the product. But he might have guessed my intent from the n=12 example, which is plain ASCII, except for the pi. Haven't we been using ? quite a while now? It's just option p on the Mac keyboard. --rwg --Your ? here came out as ? on my reader. I have a mac. When I type "option-p" into this here window, I do not get pi. I get p. I just did it. Gosper's posts often are hard for me to read, they often have had wrong line-breaks and unreadable characters combined. The solution is to post using only ASCII characters and manually placing all your line breaks exactly where they should be, using fairly short lines only --not depending on tools to insert them for you (since they may place them wrong). If you do that, everybody can read it. If you do not, some people will have trouble reading it. Your choice. <WDS ω. ⊅ε®рιи₲τοи ∫⋔ιθ ϚϘ∪∃∀⋔∫ ⊙∬⎮₣α₲εϩļ