How prove this $f_{2m+1}(x)+xf_{2m-1}(x)=0 $ only have real roots let $$f_{1}(x)=1, f_{2}(x)=1+x, f_{n}(x)=f_{n-1}(x)+xf_{n-2}(x)$$
show that
$$f_{2m+1}(x)+xf_{2m-1}(x)=0,m\in N$$only have real roots.
Thank you everyone. This problem is my frend ask me.I don't known this problem come from which books,Thank you
It is known that Hermite polynomials and Legendre polynomials only have real numbers,see: http://en.wikipedia.org/wiki/Hermite_polynomials  and
 http://en.wikipedia.org/wiki/Legendre_polynomials  But this I cann't prove it.
 A: Fisk - Polynomials, Roots, & Interlacing - Page 49 - Corollary 1.95 
Theorem. Suppose $P_1$ and $P_2$ have real interlacing zeros, then the recursion formula $$P_n=P_{n-1}+x\cdot P_{n-2}$$ defines a sequence of polynomials with only real zeros.
Note: I am stating a weaker paraphrased version to the theorem in the book.
Define $$g_m(x)=f_{m+1}(x)+x\cdot f_{m-1}(x).$$ Notice that the sequence $\{g_m\}$ includes, as a subset, the set of polynomials that you want to show have only real zeros. With this easier formula we arrive at the simple recursion: $$g_m(x)=g_{m-1}+x\cdot g_{m-2}(x).$$ Therefore, we only need to show that $\{g_m\}$ has two initial polynomials with real interlacing zeros. We calculate the first few polynomials, $$g_2(x)=3x+1 \\ g_3(x)=2x^2+4x+1 \\ g_4(x)=5x^2+5x+1 \\ g_5(x)=2x^3+9x^2+6x+1 \\ g_6(x)=7x^3+14x^2+7x+1 \\ \vdots $$ and calculate the roots, $$Roots(g_2)=\{-0.3333\} \\ Roots(g_3)=\{-0.2929,-1.7071\} \\ Roots(g_4)=\{-0.2764,-0.7236,\} \\ Roots(g_5)=\{-0.2670,-0.5000,-3.7321\} \\ Roots(g_6)=\{-0.2630,-0.4090,-1.3290\} \\ \vdots $$ We see that $g_2$ and $g_3$ have interlacing zeros, in fact this list is consecutively interlacing. Note that the consecutive interlacing was not guaranteed by the theorem; the only thing the theorem guarantees is that the remaining polynomials will all have only real zeros. Although, now that we know all the polynomials have only real zeros, there is another theorem of Fisk that tells us they will continue to consecutively interlace. 
Definition. Interlacing means that the zeros of two polynomials fit together; for example  $$a_1< b_1 < a_2 < \cdots a_n < b_n.$$ Just to be clear two polynomials can only interlace if they are within a degree of each other. 
