On 2015-01-06 12:58, Daniel Asimov wrote:

Thanks, Mike. That's a very nice paper!

--Dan

...

On Jan 6, 2015, at 12:04 PM, Mike Stay <metaweta@gmail.com> wrote:

http://arxiv.org/abs/physics/9705021

On Tue, Jan 6, 2015 at 11:16 AM, Daniel Asimov <dasimov@earthlink.net> wrote:

...

Are there extensions of Bernoulli numbers to real (or complex) index?

Is there one that is generally accepted as the natural way to do it?

It would be nice if there were a natural real-index Bernoulli function that took real values.

I read somewhere that Ramanujan used to refer to real order Bernoulli numbers in his notebooks. Does anyone know what definition he used?

--Dan

<Grumble>: a. People will never agree whether B_n is B_n(0) or B_n(1), i.e. B_1 = -1/2 or 1/2. b. In a formula, it is nearly always possible to generalize B_n to B_n(a), the Bernoulli polynomial. So change the question to "What is B_s(a)?". Obvious definition: BernoulliB[s, a]:= -s HurwitzZeta[1 - s, a] . This is numerically consistent with my occasionally repeated 3×3 product: MatrixForm[{{0, 0, BernoulliB[n, x]/m}, {0, 0, HarmonicNumber[m]/m}, {0, 0, 1}}] == Product[MatrixForm[{{(k - m)/(k + 1), (k - m)*(x + k - 1)^n/(k + 1), (x + k - 1)^n/k}, {0, (k - m)/(k + 1), 1/k}, {0, 0,1}}], {k, m}] BernoulliB[n, x] 0 0 ---------------- m HarmonicNumber[m] [0 0 -----------------] == m 0 0 1 n n k - m (k - m) (-1 + k + x) (-1 + k + x) ----- --------------------- ------------- 1 + k 1 + k k k - m 1 Product[ 0 ----- - , {k, m}] 1 + k k 0 0 1 at least for integer n and any m > n. Empirically, for noninteger n, this converges to the HurwitzZeta definition as m increases. To test this, say for m=3 In[103]:= %84 /. Product[a_, b_] :> Dot @@ (Table[a[[1]], b] /. m -> 3) /. m -> 3 During evaluation of In[103]:= Table::iterb: Iterator {k,m} does not have appropriate bounds. >> Out[103]= (0 0 1/3 BernoulliB[n,x] 0 0 11/18 0 0 1 )=={{0,0,x^n/2-1/2 (1+x)^n+1/9 (2+x)^n+1/3 (x^n/3+1/3 (1+x)^n)},{0,0,11/18},{0,0,1}} In[104]:= FullSimplify[Table[% /. MatrixForm -> Identity, {n, 0, 3}]] Out[104]= {True, True, True, False} (For {True, True, True,True}, we'd need m>3.) Using BernoulliB[n,1] = -n Zeta[1-n], m = 22 gives -n Zeta[1 - n] -> (1/232792560)(859193865 - 4262242455 2^n - 299075781161 2^n 3^(1 + n) + 2513922025 3^(2 + n) + 1877016414259 3^(1 + 2 n) - 96874882575 4^n - 613864407 4^(1 + n) 5^n + 13153780461 5^(2 + n) + 1997664536397 7^n + 77765480 3^(1 + n) 7^n - 496222352041 2^n 7^(1 + n) - 3673960371123 8^n - 19547829457 2^(2 + n) 9^n - 7235136890023 10^n + 7818300885353 11^n - 1322685 2^(3 + n) 11^n - 7110728313367 12^n + 5433803166113 13^n + 1843591886513 15^n - 803166403663 16^n + 282418051292 17^n + 16412991972 19^n) Plotting Subtract@@% {n,0,7} stays between -.01 and .003, and is of course 0 at integer n. --rwg