[math-fun] I've only now come to understand that 𝜁[s,a] ≠ HurwitzZeta[s,a],
and is instead completely batbleep: \[Zeta](s,a)==\!\(\* UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]", LimitsPositioning->True] \*SuperscriptBox[\((k + a)\), \(-s\)]\), where any term with k+a==0 is excluded. (Decipher Box Notation by pasting into Mma.) Can someone provide motivation or precedent for this definition? E.g., why not arbitrarily *define* (-n)!, for positive integer n, as Limit[k! + (-1)^n/(k + n)/Gamma@n, k -> -n] ? (See A060746 <http://oeis.org/A060746>.) Or some more useful sequence, say, Bernoulli numbers? —rwg
From the Mma doc: "HurwitzZeta[s,a] gives the Hurwitz zeta function 𝜁(s,a)." Please feel welcome to share my confusion.
http://mathworld.wolfram.com/HurwitzZetaFunction.html warns about this bizarre (un-)convention right at the start, but remains diplomatically silent about any conceivable rationale. [ One deduces that square brackets were _not_ intended to indicate old-fashioned British "floor" function; though you never know! ] WFL On 6/7/19, Bill Gosper <billgosper@gmail.com> wrote:
and is instead completely batbleep: \[Zeta](s,a)==\!\(\* UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]", LimitsPositioning->True] \*SuperscriptBox[\((k + a)\), \(-s\)]\), where any term with k+a==0 is excluded.
(Decipher Box Notation by pasting into Mma.)
Can someone provide motivation or precedent for this definition? E.g., why not arbitrarily *define* (-n)!, for positive integer n, as Limit[k! + (-1)^n/(k + n)/Gamma@n, k -> -n] ? (See A060746 <http://oeis.org/A060746>.) Or some more useful sequence, say, Bernoulli numbers? —rwg From the Mma doc: "HurwitzZeta[s,a] gives the Hurwitz zeta function 𝜁(s,a)." Please feel welcome to share my confusion. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
<< warns about this bizarre (un-)convention right at the start ... >> --- then contradicts itself a few lines later, following formula (3). Evidently we are not alone in our confusion! WFL On 6/7/19, Fred Lunnon <fred.lunnon@gmail.com> wrote:
http://mathworld.wolfram.com/HurwitzZetaFunction.html
warns about this bizarre (un-)convention right at the start, but remains diplomatically silent about any conceivable rationale. [ One deduces that square brackets were _not_ intended to indicate old-fashioned British "floor" function; though you never know! ]
WFL
On 6/7/19, Bill Gosper <billgosper@gmail.com> wrote:
and is instead completely batbleep: \[Zeta](s,a)==\!\(\* UnderoverscriptBox["\[Sum]", RowBox[{"k", "=", "0"}], "\[Infinity]", LimitsPositioning->True] \*SuperscriptBox[\((k + a)\), \(-s\)]\), where any term with k+a==0 is excluded.
(Decipher Box Notation by pasting into Mma.)
Can someone provide motivation or precedent for this definition? E.g., why not arbitrarily *define* (-n)!, for positive integer n, as Limit[k! + (-1)^n/(k + n)/Gamma@n, k -> -n] ? (See A060746 <http://oeis.org/A060746>.) Or some more useful sequence, say, Bernoulli numbers? —rwg From the Mma doc: "HurwitzZeta[s,a] gives the Hurwitz zeta function 𝜁(s,a)." Please feel welcome to share my confusion. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Bill Gosper -
Fred Lunnon