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Lemma A.4 .
Let α ∈ ℝ ∖ { 0 , − 1 , − 2 , − 3 , … } \fa\in\mathbb{R}\setminus\{0,-1,-2,-3,\ldots\} β ∈ ℝ \fb\in\mathbb{R} y ∈ ℝ y\in\mathbb{R}
(A.32) Φ ♯ ( β , α , y ) \displaystyle\Ku(\fb,\fa,y) = Φ ♯ ( β + 1 , α , y ) − y α Φ ♯ ( β + 1 , α + 1 , y ) , \displaystyle=\Ku(\fb+1,\fa,y)-\frac{y}{\fa}\Ku(\fb+1,\fa+1,y),
(A.33) Φ ♯ ( β , α , y ) \displaystyle\Ku(\fb,\fa,y) = α + y α ⋅ Φ ♯ ( β , α + 1 , y ) − α − β + 1 α ( α + 1 ) ⋅ y ⋅ Φ ♯ ( β , α + 1 , y ) . \displaystyle=\frac{\fa+y}{\fa}\cdot\Ku(\fb,\fa+1,y)-\frac{\fa-\fb+1}{\fa(\fa+1)}\cdot y\cdot\Ku(\fb,\fa+1,y).