# Tagged Questions

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The Lambert $W$ function has the property that $$W'(x) = \frac{W(x)}{x[1+W(x)]},$$ and using this one can show that its Taylor expansion about $x=a$ has the form $$W(x) = W(a) + ... 1answer 82 views ### Recurrence equation and special functions Can someone give me a proof or a hint on why the recurrence equation:$$g(k+2)=k*g(k+1)-g(k)$$has the solution:$$g(k)=c_1 {_0\tilde F_1}(;k;-1)+c_2 Y_{k-1}(2)$$where {_0\tilde F_1}(;a;x) is the ... 1answer 134 views ### How can I express such function as known functions or power series?$$\int_0^x \cfrac{1}{1+\int_0^t \cfrac{1}{2+\int_0^{t_1} \cfrac{1}{3+\int_0^{t_2} \cfrac{1}{\cdots} dt_3} dt_2} dt_1} dt =f(x)\int_{0}^{x} \frac{1}{n+h_{n+1}(t)}{d} t=h_n(x)$$... 0answers 103 views ### Perrin numbers in terms of the generalized hypergeometric function? Given the roots of x^3=x^2+1, we have sequence A001609, M(n) = x_1^n+x_2^n+x_3^n = \,_3F_2\left(\frac{-n}{3}, \frac{1-n}{3}, \frac{2-n}{3};\; \frac{1-n}{2}, \frac{2-n}{2};\; ... 1answer 180 views ### What's the reasoning for this recurrence on q-multinomial coefficients? I'm familiar with the recurrence for binomial coefficients based on Pascal's triangle. However, in general, there is the recurrence for q-multinomial coefficients given by$$ ...
I have a function defined as a set of weighted Legendre polynomials: $f(x)=\alpha_0 P_0(x) + \alpha_1 P_1(x) + \alpha_2 P_2(x) +\ldots$. I have another function similarly defined with Legendre basis ...