Oops, I've missed a bunch of opportunities to specialize parameters to
get nonugliness from that system. E.g., note the quadrinomial in the
UR k (2nd) matrix numerator:
{{{(-q^(i + k) + q^(i + j + n))/(-1 + q^(k + n)), 1}, {0, 1}},
{{(q^(i + n)*(q^j - q^(2*k))*(q^j - q^(1 + 2*k)))/((q^(i + j) - q^k)*(-1 +
q^(i + k))*
(-q^(1 + k) + q^(j + n))*(-1 + q^(k + n))),
(q^k*(q^(1 + i + j) - q^(1 + k) + q^(j + n) - q^(i + j + k + n)))/
((q^(i + j) - q^k)*(-1 + q^(i + k))*(-q^(1 + k) + q^(j + n)))}, {0, 1}}}
It will factor for j=-2i and j=2-2n (among other cases). The former leads
to
1 == (QPochhammer[-x, q]*Sum[(-1)^n*q^(((-1 + n)*n)/2)*x^n, {n, 0,
Infinity}])/
(QPochhammer[q, q]*QPochhammer[x, q]*QPochhammer[x^2, q^2]) -
Sum[(q^(1 + n)*QPochhammer[-(x/q), q, 1 + n]*QPochhammer[q*x^2, q^2, n])/
(QPochhammer[q, q, 1 + n]*QPochhammer[x, q, 1 + n]*QPochhammer[x^2, q,
n]),
{n, 0, Infinity}]
and the Lambert sum
Sum[q^n/(1 - q^(2*n)*x), {n, 0, Infinity}] ==
(QPochhammer[-q, q^2]*QPochhammer[q^4, q^4]*Sum[(-1)^n*q^n^2*x^n, {n, 0,
Infinity}])/
(QPochhammer[q, q^2]*QPochhammer[q^2/x, q^2]*QPochhammer[x, q^2,
Infinity]) -
((q + x)*Sum[(q^(1 + 2*n)*QPochhammer[-q, q^2, n]*QPochhammer[q^4, q^4,
n])/
(QPochhammer[q, q^2, 1 + n]*QPochhammer[q^2/x, q^2, 1 + n]*
QPochhammer[x, q^2, 1 + n]), {n, 0, Infinity}])/x
and their common generalization
Sum[(a^n*QPochhammer[x/a, q, n])/QPochhammer[a*x, q, n], {n, 0, Infinity}]
==
(QPochhammer[-a, q]*QPochhammer[a^2*q, q^2]*Sum[(-1)^n*q^(((-1 +
n)*n)/2)*x^n,
{n, 0, Infinity}])/(QPochhammer[a, q]*QPochhammer[(a*q)/x, q]*
QPochhammer[a*x, q]) -
(a*(q + x)*Sum[(q^n*QPochhammer[-a, q, n]*QPochhammer[a^2*q, q^2, n])/
(QPochhammer[a, q, 1 + n]*QPochhammer[(a*q)/x, q, 1 + n]*
QPochhammer[a*x, q, n]), {n, 0, Infinity}])/x
Note the "half theta series".
--rwg
On Thu, Feb 23, 2012 at 1:52 PM, Bill Gosper <billgosper(a)gmail.com> wrote:
> On Thu, Feb 16, 2012 at 12:09 PM, Bill Gosper <billgosper(a)gmail.com>wrote:
>
>> Despite the pygalgia of wrapping all my summands in HoldForms,
>
> pygodynia! The insidious bugs this causes are hard to believe.
>
>
>> I'm methodically
>> constructing all the non-ugly identities generated by the contours
>> producing Joerg's
>> Lambert series, where ugly:= trinomials or worse in a summand.
>
>
> Here's a not-too-ugly Lambert: = trinomial:
>
> Sum[q^n/(1 - a*q^n), {n, 0, Infinity}] ==
> Sum[(a^(2*k)*q^((k*(1 + 3*k))/2)*(1 + q^k*(q - a*q^(1 +
> k)))*QPochhammer[q, q, k]^2)/
> ((-1)^k*(1 + q^(1 + k))*QPochhammer[a, q, 1 + k]*QPochhammer[q, q, 1 +
> 2*k]), {k, 0, Infinity}]
>
> More generally,
>
> Sum[(j^(2*n)*q^n*QPochhammer[a/(j*q), q, n])/QPochhammer[a, q, n], {n, 0,
> Infinity}] ==
> Sum[(a^(2*k)*j^k*q^((3*(-1 + k)*k)/2)*(1 + j*q^k*(q -
> a*q^k))*QPochhammer[j*q, q, k]^2)/
> ((-1)^k*(1 + j*q^(1 + k))*QPochhammer[a, q, k]*QPochhammer[j^2*q, q, 1
> + 2*k]), {k, 0, Infinity}]
>
> This will make a nice test of the pPhiq notator I'm trying to write.
> pPhiq notation has
> even more drawbacks than pFq.
> --rwg
>
>> One of these
>> non-uglies does something novel: The base leg producing the Lambert sum
>> gets
>> multiplied by zero, leaving the somewhat peculiar identity
>>
>> Sum[t^n/QPochhammer[x, q, n], {n, 0, Infinity}] ==
>> Sum[(q^(-1 + j)*(q - x)*QPochhammer[q^(1 + j), q])/
>> (QPochhammer[q^j*t, q]*QPochhammer[q^(-1 + j)*x, q]), {j, 0,
>> Infinity}],
>>
>> whose lhs is the generating fcn of the reciprocals of (x;q)_n for
>> nonnegative
>> integer n. I quote it here on the off chance it lacks a more
>> conventional derivation.
>>
>> Linearly combining other results gives a thetalike gfcn for the
>> reciprocal pochhammers
>> 1/(x;q)_k in terms of the gfcn for the unreciprocated pochhammers (t;q)_k
>> !
>>
>> Sum[x^k*QPochhammer[t, q, k], {k, 0, Infinity}] ==
>> Sum[(q^(-(k/2) + k^2/2)*t^k*x^k)/((-1)^k*QPochhammer[x, q, 1 + k]),
>> {k, 0, Infinity}]
>> --rwg
>>
>> Note that the HoldForm workaround is fundamentally wrong because
>> HoldForm[f[n]]
>> is not a function of n !:
>>
>> In[903]:= Sum[HoldForm[f[n]], {n, 69}]
>>
>> Out[903]=69 f[n]
>>
>> I have a Sum[(-1)^(2n)*f[n],{n,Infinity}]. Morbidly, I wondered what
>> it would take to
>> punt the (-1)^(2n) automatically. Sneaking Simplify etc. into the
>> summand under the
>> HoldForm does nothing. Completely releasing the HoldForm "only" took
>> a minute or
>> so, but still wouldn't punt the (-1)^(2n). FullSimplify on the
>> unwholesome whole Sum(s)
>> has now been running for three days. It is not a large expression:
>>
>> Sum[i^n/(1 - b*q^n), {n, 0, Infinity}] ==
>> (QPochhammer[q, q]*Sum[(-1)^(2*n)*i^(-1 + n)*QPochhammer[b, q, -1 + n],
>> {n, 1, Infinity}])/(QPochhammer[q/i, q]*QPochhammer[b, q,
>> Infinity]) -
>> Sum[(q^n*QPochhammer[q, q, -1 + n])/(i*QPochhammer[b, q, n]*
>> QPochhammer[q/i, q, n]), {n, 1, Infinity}]
>>
>> On Sat, Feb 11, 2012 at 2:52 AM, Bill Gosper <billgosper(a)gmail.com>
>> wrote:
>> > On Fri, Feb 10, 2012 at 4:47 AM, Bill Gosper <billgosper(a)gmail.com>
>> wrote:
>> >> The very simple 4D matrix system for Joerg's Lambert series is
>> >> I(i, j, k, n) := [(q^(n + k - 1) * (1 - (1/(q^(k - j - i - 1))))/(1 -
>> >> q^i)), (1 - q^(n + k - 1)/(1 - q^i)); 0, 1],
>> >> J(i, j, k, n) := [(1 - (1/(q^(k - j - i - 1)))) * (1 - q^(n + j))/(1 -
>> >> (1/(q^(k - j - 1)))), - (1 - q^(n + k - 1))/(q^(k - j - 1) * (1 -
>> >> (1/(q^(k - j - 1))))); 0, 1],
>> >> K(i, j, k, n) := [(1 - q^(k - j))/((1 - q^(k - j - i)) * (1 - q^(n +
>> >> k))), - (q^(k - j - i)/(1 - q^(k - j - i))); 0, 1],
>> >> N(i, j, k, n) := [q^i * (1 - q^(n + j))/(1 - q^(n + k)), 1; 0, 1]
>> >>
>> >> Prod(N(t,x,x+1,n),n,0,oo) computes the Lambert series.
>> >> The contour in the n-i plane, (t,x,x+1,0)...(t,x,x+1,oo)
>> ...(oo,x,x+1,oo)
>> >> = (t,x,x+1,0)...(oo,x x+1,0) ...(oo,x,x+1,oo)
>> >> directly computes Joerg's symmetry observation:
>> >>
>> >> sum(t^n/(1 - q^n * x),n,0,inf) = sum(x^i/(1 - q^i * t),i,0,inf)
>> >> --rwg
>> > Aggrieved at not finding Joerg's Theta-convergent, Pochhammer-free
>> identity,
>> > I exhaustively searched all computationally feasible coordinate changes
>> of
>> > this 4D system. The winner was i=t-1, j=-1, leaving the extremely
>> simple
>> > 2D system
>> > [ n + 2 k + i - 1 ]
>> > [ n + 2 k + i 1 - q ]
>> > [ q ----------------------------- ]
>> > [km(k, n) := [ k + i n + k - 1 ],
>> > [ (1 - q ) (1 - q ) ]
>> > [ ]
>> > [ 0 1 ]
>> >
>> > [ k + i 1 ]
>> > [ q -------------- ]
>> > nm(k, n) := [ n + k - 1 ]]
>> > [ 1 - q ]
>> > [ ]
>> > [ 0 1 ]
>> > (the specialization of j to -1 permits "sidestepping" to almost pure sum
>> > notation, ruling out Pochhammers from the contour.)
>> > Running these through MProd, Julian's new matrix product to sum
>> converter,
>> >
>> > {MProd[{{q^(i + 1), 1/(1 - q^n)}, {0, 1}}, {n, x, Infinity}] ,
>> > MProd[{{0, -(1/(q^(k + i) - 1))}, {0, 1}}, {k, 1, Infinity}]} ->
>> > {MProd[{{q^(x + 2*k + i), (1 -
>> > q^(x + 2*k + i - 1))/((1 - q^(k + i))*(1 -
>> > q^(x + k - 1)))}, {0, 1}}, {k, 1, Infinity}] , {{0, 1}, {0,
>> 1}}}
>> >
>> > elicited from Mma a bunch of bogus nonconvergence complaints, then a
>> > bunch of infectious and gratuitous "Indeterminate"s, and ultimately an
>> > utterly useless and incorrect "False". Changing Equal to Rule and Dot
>> to List,
>> > and then changing them back:
>> > In[101]:= Dot @@ # & /@ (% /. Indeterminate -> 0) /. Rule -> Equal
>> >
>> > Out[101]= {{0, Sum[(q^(1 + i))^(k39 - x)/(1 - q^k39), {k39, x,
>> > Infinity}]}, {0, 1}} ==
>> > {{0, Sum[(q^((-1 + k43)*(i + k43 + x))*(1 -
>> > q^(-1 + i + 2*k43 + x)))/((1 - q^(i + k43))*(1 -
>> > q^(-1 + k43 + x))), {k43, 1, Infinity}]}, {0, 1}}
>> >
>> > At last!
>> > --rwg
>> > So now we know that it's possible to exclude Pochhammers from a system
>> > before choosing the contour. Further recoordinatizations of that 2D
>> system
>> > are guaranteed to remain Pochhammer free. Other integer j should also
>> work,
>> > but probably won't yield anything we couldn't get from Joerg's identity
>> plus
>> > partial fractions. Path invariance is preserved by differentiating
>> > and integrating
>> > wrt to variables not appearing in the upper left [1,1] element, but
>> this is
>> > unlikely to yield anything we couldn't get from differentiating and
>> integrating
>> > the actual sums.
>> >
>> > (This is weird: Firefox will *only* work in virtual XP when I'm home,
>> and
>> > *only* work in the Lion OS at the Tastebuds restaurant. What will
>> happen
>> > in Zieglerville?)
>> >
>> >> While Mma 8.04 knows the q-binomial and q-exponential sums, it appears
>> not
>> >> to know the q-Gauss (Heine) nor q-Dixon, e.g.
>> >> QHypergeometricPFQ[{a, (-Sqrt[a])*q, b, c}, {-Sqrt[a], (a*q)/b,
>> >> (a*q)/c}, q, (Sqrt[a]*q)/(b*c)] ==
>> >> (QPochhammer[a*q, q]* QPochhammer[(Sqrt[a]*q)/b, q]*
>> >> QPochhammer[(Sqrt[a]*q)/c, q]*QPochhammer[(a*q)/(b*c), q])/
>> >> (QPochhammer[Sqrt[a]*q, q]*QPochhammer[(a*q)/b, q]*
>> >> QPochhammer[(a*q)/c, q]* QPochhammer[(Sqrt[a]*q)/(b*c), q])
>> >>
>> >> It can, however, test this symbolically by expanding at q=0.
>> >>
>> >> On Wed, Feb 8, 2012 at 12:01 AM, Bill Gosper <billgosper(a)gmail.com>
>> wrote:
>> >>> Puzzle: Assuming[0 < q < 1,
>> >>> Limit[QPochhammer[a*c, q]/QPochhammer[b*c, q], c -> \[Infinity]]]
>> >>>
>> >>> Or maybe Assuming[0 < q < 1,
>> >>> Limit[QPochhammer[a*x, q]*
>> >>> QPochhammer[b*x, q]/QPochhammer[c*x, q]/QPochhammer[a*b*x/c, q],
>> >>> x -> \[Infinity]]]
>> >>>
>> >>> Back to Joerg's sum. This time I got
>> >>> Sum[t^n/(1 - q^n*x), {n, 0, Infinity}] ==
>> >>> (QPochhammer[q, q]*Sum[t^n*QPochhammer[x, q, n], {n, 0, Infinity}])/
>> >>> (QPochhammer[q/t, q]*QPochhammer[x, q]) -
>> >>> Sum[(q^k*QPochhammer[q, q, k - 1])/(QPochhammer[q/t, q, k]*
>> >>> QPochhammer[x, q, k]),
>> >>> {k, 1, Infinity}]/t
>> >>>
>> >>> Note the middle sum is the g.f of the finite qpochhammers of two fixed
>> >>> arguments.
>> >>> Shouldn't that sum be easy? Is it in Fine? I left my BHS at Neil's.
>> >>> (But I can never
>> >>> find stuff in there anyway.)
>> >>>
>> >>> Trying the g.f. sum, I hit this weirdy:
>> >>> Sum[(c^n*QPochhammer[b, q, n])/(b^n*QPochhammer[c*q, q, n]), {n, 0,
>> >>> Infinity}] ==
>> >>> (b*(-1 + c))/(-b + c)
>> >>>
>> >>> independent of q. Plotting for b=2/3,c=1/3, it holds somewhat past
>> >>> q=1, and then
>> >>> develops a very interesting collection of poles.
>> >>> --rwg
>> >>>
>> >>> On Mon, Feb 6, 2012 at 5:46 PM, Bill Gosper <billgosper(a)gmail.com>
>> wrote:
>> >>>> Joerg>From the department of formulas that could possibly
>> >>>> be found in some paper from 200 years ago.
>> >>>>
>> >>>>>Semi-recently I made the observation (and posted it here) that
>> >>>> sum(n>=0, x*q^n/(1-x*q^n) can be computed as
>> >>>> sum(n>=0, q^(n^2+n)*x^(n+1)*(1 - x*q^(2*n+1)) / ((1 - x*q^n)*(1 -
>> q*q^n)))
>> >>>> and this allows the fast computation of sums of inverse Fibs
>> >>>> (and infinitely more sums of reciprocal order-two linrecs)
>> >>>> without splitting into even and odd part (yes, RWG, AIM304 exists).
>> >>>>
>> >>>> YOW, I'd forgotten that stunt!
>> >>>>
>> >>>>>After much effort I found that
>> >>>> sum(n>=0, t^n / (1-x*q^n) )
>> >>>>
>> >>>> Random aside: Per my recent (generalized) contiguous Heine mail,
>> >>>> with c=a and d=b q = x q and z=t,
>> >>>> sum(((t^n)/(1 - q^n * x)),n,0,inf) =
>> >>>> ((a * (1 - q * t) * x * sum(((q^(2 * n) * t^n)/(1 - q^n *
>> x)),n,0,inf)
>> >>>> - ( - q * t * x + q * x - a * q * t + a) * sum(((q^n * t^n)/(1 - q^n
>> >>>> * x)),n,0,inf) - q + a)/(q * (t - 1)))
>> >>>>
>> >>>> independent of a!
>> >>>>
>> >>>> Joerg>can be computed as
>> >>>> sum(n=0, S, (1-x*t*q^(2*n))*(x*t)^n*q^(n^2) / (
>> (1-x*q^n)*(1-t*q^n)) )
>> >>>> (which btw. is symmetric in x and t).
>> >>>>
>> >>>> This is driving me nuts. The 3x3 system mentioned in that Heine
>> mail refuses
>> >>>> to triangularize for this Lambert case, which is degenerate enough
>> to permit
>> >>>> discarding two parameters from the Heine N matrix. One of these,
>> call it k, can
>> >>>> be via a reversible (unit determinant) "recoord" (our new name for
>> >>>> "transformal").
>> >>>> But k then persists in the other matrices, and K, the k-bumpng
>> matrix, does not
>> >>>> become the 3x3 identity! Thus, wandering around in the k-n plane is
>> a no-op,
>> >>>> because, by path invariance, N(k,n).K(k,n+1) = K(k,n).N(k+1,n), but
>> >>>> N(k,n)=N(k+1,n)! There must be some use for this.
>> >>>>
>> >>>> Rolling back to a 1998 Heine 2x2 system (that was born triangular),
>> then
>> >>>> replacing n+1 by n+h by shifting i,j,k,n by 1-h,1-h,1-h,h-1, then
>> >>>> specializing h to i*t (and discarding the I matrix), then q^t->t,
>> >>>> gives the J,K,N matrices
>> >>>>
>> >>>> {{{-(((1 - q^(n + j))*(1 - 1/(q^(k - j - 2)*t))*t)/(q^(n + j + 1)*(1
>> -
>> >>>> 1/q^(k - j - 1)))),
>> >>>> -((1 - q^(n + k - 1))/ (q^(n + k - 1)*(1 - 1/q^(k - j - 1))))}, {0,
>> 1}},
>> >>>> {{-(((q^(n + k - 1)*(1 - q^(k - j))*t)/(1 - q^(n + k)))*(1 - q^(k -
>> >>>> j - 1)*t)),
>> >>>> 1/(1 - q^(k - j - 1)*t)}, {0, 1}},
>> >>>> {{(q^(k - j - 1)*(1 - q^(n + j))*t)/(1 - q^(n + k)), 1}, {0, 1}}}
>> >>>>
>> >>>> Then pegging j at x and closing a rectangle based at k=x+1,n=0, then
>> >>>> q^x->x gives a *different* theta-convergent series symmetric in t
>> and x:
>> >>>>
>> >>>> Sum[t^n/(1 - q^n*x), {n, 0, Infinity}] ==
>> >>>> Sum[((-1)^k*q^(k^2/2 - k/2)*QPochhammer[q, q, k]*t^k*x^k)/
>> >>>> (QPochhammer[t, q, k + 1]*QPochhammer[x, q, k + 1]), {k, 0,
>> >>>> Infinity}]
>> >>>>
>> >>>> (Dialog) In[649]:= Simplify[
>> >>>> FunctionExpand[Series[%[[2]] /. \[Infinity] -> 6, {t, 0, 6}]]]
>> >>>>
>> >>>> (Dialog) Out[649]= SeriesData[t, 0, {(1 - x)^(-1), (1 - q x)^(-1), (
>> >>>> 1 - q^2 x)^(-1), (1 - q^3 x)^(-1), (1 - q^4 x)^(-1), (
>> >>>> 1 - q^5 x)^(-1), (1 - q^6 x)^(-1)}, 0, 7, 1]
>> >>>>
>> >>>> Like I said: nuts. I'll bet yours|Fine's is in there somewhere.
>> >>>> --rwg
>> >>>>
>> >>>>>Now this one turns out to be easily obtained by setting a := -b
>> >>>> in Fine's "versatile" relation (14-1) (p.15 in "Basic Hypergeometric
>> >>>> Series and Applications"). Anybody with one bit of interest in
>> q-things:
>> >>>>
>> >>>> Buy this book, it's a marvel.
>>
>
>
