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Mathematica provides the MathieuC[a,q,z] and MathieuS[a,q,z] functions - as well as some other Mathieu-related functions. Maple does similarly. But in addition, Maple provides a MathieuFloquet[a,q,z] function.

The point is that (contrary to the Mathematica documentation) MathieuC and MathieuS do not necessarily individually satisfy the Floquet periodicity condition. But a linear combination of the two will.

The Maple documentation shows how MathieuC[a,q,z] and MathieuS[a,q,z] may be found from (linear combinations of) MathieuFloquet[a,q,z] and MathieuFloquet[a,q,-z].

In Mathematica, I want to do the opposite; I want to get MathieuFloquet[a,q,z] in terms of MathieuC[a,q,z] and MathieuS[a,q,z].

Sounds straightforward, but I can't see how to do it.

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    $\begingroup$ What is the definition of MathieuFloquet[a,q,z] in terms of the other functions? $\endgroup$
    – MarcoB
    Dec 1, 2023 at 14:56

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Up to Abramowitz, M., and Stegun, I., eds. Handbook of Mathematical Functions. New York: Dover Publications and Maple online help, the following relations are valid

MathieuC[a, q,  x] == ( MathieuFloquet[a, q, x] + MathieuFloquet[a, q, -x])/(2*
 MathieuFloquet[a, q, 0]) && MathieuS[a, q, x] == (MathieuFloquet[a, q, x] - MathieuFloquet[a, q, -x])/(2*
 MathieuFloquetPrime[a, q, 0])

Then

Solve[%, {MathieuFloquet[a, q, x],  MathieuFloquet[a, q, -x]}]

{{MathieuFloquet[a, q, x] -> MathieuC[a, q, x] MathieuFloquet[a, q, 0] + MathieuFloquetPrime[a, q, 0] MathieuS[a, q, x], MathieuFloquet[a, q, -x] -> MathieuC[a, q, x] MathieuFloquet[a, q, 0] - MathieuFloquetPrime[a, q, 0] MathieuS[a, q, x]}}

does the job. Unfortunately, I don't know formulas for MathieuFloquetPrime[a, q, 0] and MathieuFloquet[a, q, 0].

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