Question

For any integer $n\ge 3$, let $P(x)=\sum\limits_{i(=2k)\ge 0}^{n}\binom{n}{2k}(1-x)^k$, $Q(x)=\sum\limits_{i(=2k+1)\ge 0}^{n}\binom{n}{2k+1}(1-x)^{k+1}$. Define $\frac{P(x)}{Q(x)}\equiv\sum\limits_{i=0}^{\infty}c_ix^i $. I like to know $c_0> c_1> c_2>\cdots$, but I don't know how to show it.

Answer 1

The polynomials $P_n$ and $Q_n$ can be written as $$ P_n(x)=(1+\sqrt{1-x})^n+(1-\sqrt{1-x})^n, \qquad Q_n(x)=\sqrt{1-x}\bigl((1+\sqrt{1-x})^n-(1-\sqrt{1-x})^n\bigr). $$ In particular, they are both solutions to the linear difference equation $P_{n+1}(x)=2P_n(x)-xP_{n-1}(x)$ implying that their quotient $f_n(x)=P_n(x)/Q_n(x)$ satisfies the nonlinear recursion $$ f_{n+1}(x)=\frac{1+f_n(x)}{1+(1-x)f_n(x)} $$ (already observed experimentally by Martin Rubey).

The property saying that the coefficients of the power series $f(x)=c_0+c_1x+c_2x^2+\dots$ satisfy $c_0\ge c_1\ge c_2\ge\dots$ can be rephrased as $c_0-(1-x)f(x)\ge0$, where the record $a_0+a_1x+a_2x^2+\dots\ge0$ means that all $a_i\ge0$. This makes the original problem equivalent to $1-(1-x)f_n(x)\ge0$ for all $n$. This follows by induction on $n$. The base of induction is clear verification (already done by the author for a few first $n$). Assuming that it is true for a given $n$ and using that the non-negative expansion $1-(1-x)f_n(x)$ has the vanishing constant term, we conclude that the (formal) power series $$ \frac1{1-\frac12\bigl(1-(1-x)f_n(x)\bigr)} =1+\sum_{k=1}^\infty\frac1{2^k}\bigl(1-(1-x)f_n(x)\bigr)^k $$ is non-negative. It remains to apply the above recursion for $f_n(x)$: $$ 1-(1-x)f_{n+1}(x) =\frac x{1+(1-x)f_n(x)} =\frac x2\cdot\frac1{1-\frac12\bigl(1-(1-x)f_n(x)\bigr)}, $$ and the desired property follows.

Answer 2

(Not an answer - but too large for a comment) here are recurrences for the first few n. I don't have time right now to look at them, but shouldn't be hard to spot a pattern...

(1) -> P n == reduce(+, [binomial(n, 2*k)*(1-x)^k for k in 0..n | 2*k Q n == reduce(+, [binomial(n, 2*k+1)*(1-x)^(k+1) for k in 0..n | 2*k+1 n:=3; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 10))

   (3)  [[f(n): - 4f(n + 1) + f(n) + 1= 0,f(0)= 1]]

(4) -> n:=4; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 10))

   (4)  [[f(n): - 8n f(n + 1) + 4n f(n) + n= 0,f(0)= 1]]

(5) -> n:=5; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 30))

                                                                      1
   (5)  [[f(n): - 16f(n + 2) + 12f(n + 1) - f(n) + 1= 0,f(0)= 1,f(1)= -]]
                                                                      2

(6) -> n:=6; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 30))

   (6)
                                                                        1
   [[f(n): - 32n f(n + 2) + 32n f(n + 1) - 6n f(n) + n= 0,f(0)= 1,f(1)= -]]
                                                                        2

(7) -> n:=7; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 30))

   (7)
   [
     [f(n): 64f(n + 3) - 80f(n + 2) + 24f(n + 1) - f(n) - 1= 0, f(0)= 1,
            1        3
      f(1)= -, f(2)= -]
            2        8
     ]

(8) -> n:=8; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))

   (8)
   [
     [f(n): 128n f(n + 3) - 192n f(n + 2) + 80n f(n + 1) - 8n f(n) - n= 0,
                     1        3
      f(0)= 1, f(1)= -, f(2)= -]
                     2        8
     ]

(9) -> n:=9; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))

   (9)
   [
     [f(n): - 256f(n + 4) + 448f(n + 3) - 240f(n + 2) + 40f(n + 1) - f(n) + 1= 0
       ,
                     1        3         5
      f(0)= 1, f(1)= -, f(2)= -, f(3)= --]
                     2        8        16
     ]

(10) -> n:=10; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))

   (10)
   [
     [
       f(n):
           - 512n f(n + 4) + 1024n f(n + 3) - 672n f(n + 2) + 160n f(n + 1)
         + 
           - 10n f(n) + n
           =
           0
       ,
                     1        3         5
      f(0)= 1, f(1)= -, f(2)= -, f(3)= --]
                     2        8        16
     ]

(11) -> n:=11; guessPRec(entries complete first(coefficients series(P n/Q n, x=0), 100))

   (11)
   [
     [
       f(n):
           1024f(n + 5) - 2304f(n + 4) + 1792f(n + 3) - 560f(n + 2) + 60f(n + 1)
         + 
           - f(n) - 1
           =
           0
       ,
                     1        3         5
      f(0)= 1, f(1)= -, f(2)= -, f(3)= --]
                     2        8        16
     ]

and, in case it helps, here is the (nonlinear) recurrence for the sequence of $P_n/Q_n$:

(19) -> guessRec([P n/Q n for n in 1..100])

                                                                     1
   (19)  [[f(n): ((- x + 1)f(n) + 1)f(n + 1) - f(n) - 1= 0,f(0)= - -----]]
                                                                   x - 1