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A double integral average of xᵗ with two rows and two columns has eight quadratic transformations into itself, each with two free parameters and two independent variables. Three of the transformations change double hypergeometric series of order two into series of order three, and one of these (modified by a linear transformation applicable to polynomial cases) permits a very direct proof of the addition theorem and related results for Gegenbauer polynomials. Two others transform Appell's F₂ into F₂ or F₁, and two transform F₁ into F₁. One of the latter contains Landen's transformation of the first and second incomplete elliptic integrals, and the other contains Bartky's transformation of the third complete elliptic integral.
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B. C. Carlson (1976) studied this question.
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