Skip to main content

bugs - Taylor expansion of a function containing QPochhammer[q, q, n]


I want to get the following series expansion:


Series[QPochhammer[q, q, 3], {q, 0,4}]


but in Mathematica 11.0, I obtain the following gibberish:


bad output


There weren't any problems of this kind in Mathematica 8.0.


The following example with the infinite product


Series[QPochhammer[q, q], {q, 0, 4}]

works perfectly fine, producing the series


good result


Documentation regarding QPochhammer can be found here.




Q: What causes this problem with finite products and how to avoid it?




Answer



The reason it doesn't work is because somewhere between v10.0 and v10.3, the seventh definition of QPochhammer was modified to check that the second argument does not depend on the expansion parameter.


To restore previous behavior, clear the seventh definition,


(*Triggers loading of definitions*)
Series[QPochhammer[0, x, 3], {x, 0, 1}];

(*Unprotect*)
Unprotect[QPochhammer];


QPochhammer /: 
 System`Private`InternalSeries[
   HoldPattern[QPochhammer][System`SeriesDump`w_, 
    System`SeriesDump`q_, 
    System`SeriesDump`k_Integer?Positive], {System`SeriesDump`z_, 
    System`SeriesDump`p_, System`SeriesDump`n_Integer}] /; 
 Internal`DependsOnQ[System`SeriesDump`w, System`SeriesDump`z] && ! 
    Internal`DependsOnQ[{System`SeriesDump`q, System`SeriesDump`p}, 
    System`SeriesDump`z] =.


Then add the definition from earlier versions,


QPochhammer /: 
 System`Private`InternalSeries[
   HoldPattern[QPochhammer][System`SeriesDump`w_, 
    System`SeriesDump`q_, 
    System`SeriesDump`k_Integer?Positive], {System`SeriesDump`z_, 
    System`SeriesDump`p_, System`SeriesDump`n_Integer}] /; 
  Internal`DependsOnQ[System`SeriesDump`w, System`SeriesDump`z] := 
 Module[{System`SeriesDump`lim, System`SeriesDump`ord, 

   System`SeriesDump`qq, System`SeriesDump`ww}, 
  System`SeriesDump`lim = 
   System`SeriesDump`getExpansionPoint[System`SeriesDump`w, 
    System`SeriesDump`z, 
    System`SeriesDump`p]; (System`SeriesDump`ord = 
     Min[System`SeriesDump`k, 
      System`SeriesDump`AdjustExpansionOrder[System`SeriesDump`w, 
       System`SeriesDump`lim, System`SeriesDump`z, 
       System`SeriesDump`p, System`SeriesDump`n]]; 
    System`SeriesDump`ww = 

     System`Private`InternalSeries[
      System`SeriesDump`w, {System`SeriesDump`z, System`SeriesDump`p, 
       System`SeriesDump`n}]; 
    System`SeriesDump`qq = 
     System`Private`InternalSeries[
      System`SeriesDump`q, {System`SeriesDump`z, System`SeriesDump`p, 
       System`SeriesDump`n}]; 
    1 + Plus @@ 
      Table[(-System`SeriesDump`ww)^
        System`SeriesDump`m System`SeriesDump`qq^

        Binomial[System`SeriesDump`m, 2] QBinomial[
         System`SeriesDump`k, System`SeriesDump`m, 
         System`SeriesDump`qq], {System`SeriesDump`m, 
        System`SeriesDump`ord}]) /; System`SeriesDump`lim === 0];

Protect[QPochhammer];

Now we get the desired behavior,


Series[QPochhammer[q, q, 3], {q, 0, 4}]



Desired result



Warning The seventh definition was modified in later releases probably because someone discovered that it leads to incorrect results in some cases. Proceed with caution.


Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more