I'm performing a close critical study of David Nelson's Penguin Dictionary of Mathematics (4th ed., 2008).
Under the entry hypergeometric differential equation, it suggests the form: $$x (1 - x) \dfrac {\mathrm d^2 \phi} {\mathrm d x^2} + [c - (a + b - 1) x] \dfrac {\mathrm d \phi} {\mathrm d x} - a b \phi = 0$$
However, everywhere else I look, I see it defined as: $$x (1 - x) \dfrac {\mathrm d^2 \phi} {\mathrm d x^2} + [c - (a + b + 1) x] \dfrac {\mathrm d \phi} {\mathrm d x} - a b \phi = 0$$
I suspect, but would like to be certain, that Nelson's presentation is in fact incorrect, or whether it's a variant format which is equally acceptable as a representational format.
Before I report on this as an acual error, can it be confirmed that it is in fact wrong?
Any homogenous linear equation of second order with singular points $(0,1,\infty)$ $$x(1-x)f'' + (a + b x) f'[x] +c f[x]=0$$ is a hypergeometric differential equation, the regular solution at (0,1) being the hypergemetric series $$\, _2F_1\left(-\frac{1}{2} \sqrt{b^2+2 b+4 c+1}-\frac{b}{2}-\frac{1}{2},\frac{1}{2} \sqrt{b^2+2 b+4 c+1}-\frac{b}{2}-\frac{1}{2};a;x\right).$$
Only the second standard form $$f'(x) (c-x (a+b+1))-a b f(x)+(1-x) x f''(x)=0$$ yields the simple form of the parameters $$\, _2F_1\left(a,b,c,x\right)$$ that generalizes the geometric series
$$\sum_n \frac{a_n b_n}{c_n n!} \ x^n = 1+\frac{a b x}{c}+\frac{a (a+1) b (b+1) x^2}{2 c (c+1)}+\frac{a (a+1) (a+2) b (b+1) (b+2) x^3}{6 c (c+1) (c+2)}+O\left(x^4\right)$$