24. Calculate coefficients of contracted feedback
All examples · Glossary: coefficient recurrence · Papers, proofs, Lean, and code
The Special Functions monograph places contracted arguments inside a nonlinear differential equation. We will calculate the first coefficients for its scales \(q=1/3\) and \(r=2/3\).
Read the defining equation
The normalized odd series satisfies
\[T'(z)=1+T(qz)T(rz),\qquad T(0)=0.
\]
The prime denotes differentiation with respect to \(z\). Write
\[T(z)=z+c_1z^3+c_2z^5+c_3z^7+\cdots.
\]
The coefficient of \(z\) is one because the defining equation gives \(T'(0)=1\). Substitution scales each coefficient by its degree:
\[T(qz)=qz+c_1q^3z^3+c_2q^5z^5+\cdots.
\]
Match the first two nonconstant derivative coefficients
The coefficient of \(z^2\) in \(T(qz)T(rz)\) is \(qr\). Since differentiation turns \(c_1z^3\) into \(3c_1z^2\),
\[3c_1=qr=\frac29,\qquad c_1=\frac2{27}.
\]
The coefficient of \(z^4\) has two contributions, from degrees one and three:
\[5c_2=c_1(qr^3+q^3r)
=\frac2{27}\left(\frac8{81}+\frac2{81}\right)
=\frac{20}{2187}.
\]
Thus \(c_2=4/2187\), and the polynomial through degree five is
\[T_{\le5}(z)=z+\frac2{27}z^3+\frac4{2187}z^5.
\]
For instance, its value at one is exactly \(2353/2187\). This is the value of the displayed polynomial. An error bound for its difference from the full function requires a bound on the omitted tail.
Write the recurrence
Set \(c_0=1\). Matching degree \(2n\) gives
\[c_n=\frac1{2n+1}\sum_{i=0}^{n-1}
q^{2i+1}r^{2(n-1-i)+1}c_i c_{n-1-i},\qquad n\ge1.
\]
Each new coefficient depends on finitely many earlier ones. For \(n=3\), the three products give
\[7c_3=c_2(qr^5+q^5r)+c_1^2q^3r^3
=\frac{136}{1594323}+\frac{32}{531441}
=\frac{232}{1594323}.
\]
Hence \(c_3=232/11160261\), agreeing with the monograph's displayed member.
Try it
Use the same recurrence with \(q=r=1/2\). Find \(c_1\) and \(c_2\).
Answer: \(c_1=(1/4)/3=1/12\). Then \(5c_2=(1/12)(1/16+1/16)=1/96\), so \(c_2=1/480\).
These finite coefficient identities explain the construction. The monograph's analytic existence, growth and separation results require their full arguments.
Papers, proofs, Lean, and code
Terms in this example: Derivative in the series example · Differential equation · Initial condition · Contracted argument · Coefficient recurrence
This example expands the defining coefficient recurrence for a member of Special Functions. Finite coefficient checks do not replace the monograph’s analytic existence, growth or separation arguments.
Special Functions
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Source passages:
Edition guide and reproduction instructions. Use each work’s Code and setup link for its implementation. The arithmetic check below covers this tutorial separately.
Python arithmetic check for this example: contracted_feedback(). This checks the displayed rational, modular or finite-set calculations exactly.
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