The hypergeometric equation and about its history. “The Hypergeometric equation is known as Gauss’s hypergeometric ordinary differential equation. Which is represented as x ( x − 1 ) y ″ + [ c − ( a + b + 1 ) x ] y ′ − a b y = 0 . Where a , b , c are constant. These polynomials are written so that the series solution takes a most practical Form y 1 ( x ) = 1 + a b 1 ! c x + a ( a + 1 ) b ( b + 1 ) 2 ! c ( c + 1 ) x 2 + ⋅ ⋅ ⋅ . This series is called the hypergeometric series and its sum y 1 ( x ) is called hypergeometric function.” The term hypergeometric series was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Leonhard Euler studied hypergeometric series but Carl Friedrich Gauss (1813) provided the first full systematic treatment. Studies of the nineteenth century said those of Ernst Kummer (1836), and the basic narrative of the hypergeometric function by Bernhard Riemann (1857) by means of the differential equation that he satisfies.

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