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  • https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/04%3A_Series_Solutions/4.08%3A_Hypergeometric_Functions
    Hypergeometric functions are probably the most useful, but least understood, class of functions. They typically do not make it into the undergraduate curriculum and seldom in graduate curriculum. Most...Hypergeometric functions are probably the most useful, but least understood, class of functions. They typically do not make it into the undergraduate curriculum and seldom in graduate curriculum. Most functions that you know can be expressed using hypergeometric functions.
  • https://math.libretexts.org/Bookshelves/Differential_Equations/A_Second_Course_in_Ordinary_Differential_Equations%3A_Dynamical_Systems_and_Boundary_Value_Problems_(Herman)/07%3A_Special_Functions/7.05%3A_Hypergeometric_Functions
    \[{ }_{2} F_{1}(\alpha, \beta ; \gamma ; x)=\sum_{n=0}^{\infty} \dfrac{(\alpha)_{n}(\beta)_{n}}{n !(\gamma)_{n}} x^{n} . \nonumber \] \[y(x)=A_{2} F_{1}(\alpha, \beta ; \gamma ; x)+B_{2} x_{2}^{1-\gam...\[{ }_{2} F_{1}(\alpha, \beta ; \gamma ; x)=\sum_{n=0}^{\infty} \dfrac{(\alpha)_{n}(\beta)_{n}}{n !(\gamma)_{n}} x^{n} . \nonumber \] \[y(x)=A_{2} F_{1}(\alpha, \beta ; \gamma ; x)+B_{2} x_{2}^{1-\gamma} F_{1}(1-\gamma+\alpha, 1-\gamma+\beta ; 2-\gamma ; x) . \nonumber \] \[{ }_{1} F_{1}(\alpha ; \gamma ; u)=\lim _{\beta \rightarrow \infty}{ }_{2} F_{1}\left(\alpha, \beta ; \gamma ; \dfrac{u}{\beta}\right) . \nonumber \]

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