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Apply q-Functions in Discrete Calculus
Mathematica 7 makes extensive use of q-functions in all discrete calculus operations.
In[1]:=

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problems = {HoldForm[\!\(TraditionalForm\`\*

TemplateBox[{TemplateBox[{"k", "q"}, "QGamma"],"k"},

"DifferenceDelta2"]\)], HoldForm[\!\(TraditionalForm\`\*

TemplateBox[{TemplateBox[{"k", "n", "q"}, "QBinomial"],"k"},

"DiscreteRatio2"]\)], HoldForm[\!\(TraditionalForm\`

\*UnderoverscriptBox[\(\[Product]\), \(k = 0\), \(n - 1\)]\((1 - a\ 

\*SuperscriptBox[\(q\), \(k\)])\)\)], HoldForm[\!\(TraditionalForm\`

\*UnderscriptBox[\(\[Product]\), \(k\)]sinh(k + 1)\)], 

   HoldForm[\!\(TraditionalForm\`

\*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)]

\*FractionBox[\(1\), \(

\*SuperscriptBox[\(2\), \(k\)] + 1\)]\)], 

   HoldForm[\!\(TraditionalForm\`

\*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)]\*

FractionBox[

RowBox[{

SuperscriptBox["z", "k"], " ", 

TemplateBox[{"a","q","k"},

"QPochhammer"], " ", 

TemplateBox[{"b","q","k"},

"QPochhammer"]}], 

RowBox[{

TemplateBox[{"q","q","k"},

"QPochhammer"], " ", 

TemplateBox[{"c","q","k"},

"QPochhammer"]}]]\)]};
In[2]:=

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FormulaGallery[forms_List] := 

 Module[{vals = ParallelMap[ReleaseHold, forms]}, 

  Text@TraditionalForm@

    Grid[Table[{forms[[i]], "==", vals[[i]]}, {i, Length[forms]}], 

     Dividers -> {{True, False, False, True}, All}, 

     Alignment -> {{Right, Center, Left}, Baseline}, 

     Background -> LightYellow, Spacings -> {{4, {}, 4}, 1}]]
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FormulaGallery[problems]
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