="Dan Asimov" <dasimov@earthlink.net> Wikipedia calls the original def. "First" Bernoulli numbers and the version with B(1) = 1/2 "Second" Bernoulli numbers.
Oh my! Surely these should be "Zeroth" Bernoulli numbers and "First" Bernoulli numbers respectively? Wikipedia aside, the heat at the core of this dispute seems to be over nothing more than the assignment of the title "THE Bernoulli numbers". Since Lushny presented this as a contest, perhaps B^(1) really ought to be crowned as "THE WORLD CHAMPION Bernoulli numbers"? It's a superpower war over control of the tiny notational duchy consisting of the letter "B"! This is contentious only because of the scarcity of single letters (hence the trend in computing to eschew these for anything other than trivial temporary indices, and to use longer more descriptive identifiers for things with any significant persistence or scope). Maybe we should base mathematical notation using Egyptian hieroglyphs or Chinese ideograms instead of alphabetic characters--then maybe these kinds of arguments over notation wouldn't arise? While I personally found Lushny's reasons to prefer B^(1) persuasive, I found it telling that throughout it all he could keep the discussion quite lucid merely by using the notations B^(0) and B^(1). But if you really want to reduce the clutter in your public expositions why not simply say something like: "Given the generating function for the Bernoulli polynomials B^(x)(z) = <RHS eq. (1)>, we see that when x=1 we have B^(1)(z) = <RHS eq. (3)>. Below we will write the coefficients B_n(1) simply as B_n. [* footnote: Beware! Some Authors prefer to write B_n for B_n(0) instead]" ? This seems a small tax to pay while Ye Olde Misguyded Stylinge withers away. --MLB