This question is a continuation of this post.
Currently I am reading the paper Excited Random Walk in One Dimension.
At page $8$ right column, the authors obtain equation $(33)$.
The joint probability of eating $r$ consecutive cookies on the right and then $l$ consecutive cookies on the left before the next right cookie is eaten, under the initial condition that the walk starts at the position of the right extreme cookie, is $$P(r,l) = 4pq \frac{\Gamma(L)}{\Gamma(L-2q)} \frac{\Gamma(L+r-1-2q)}{\Gamma(L+r-2p)}\frac{\Gamma(L+l+r-1-2p)}{\Gamma(L+l+r)}.$$
Let $s = r+l.$ Then the authors used the following formula $$\bigg\langle \frac{r}{s} \bigg\rangle = \int_0^\infty ds \int_0^s \frac{r}{r+l} \, \tilde{P}(r,s)\,dr$$ to calculate the average ratio for the number of right cookies to total cookies eaten in a single cycle I am confused with the upper limit of the second integral. Why do we integrate till $s$ instead of $\infty?$ On the other hand,
Question: What is the formula to calculate average ratio for the number of right cookies to total cookies eaten in a single cycle?
Is it $$\sum_{l=1}^\infty \sum_{r=1}^\infty \frac{r}{l+r} P(r,l)?$$
a) Changing the variables from $(r,\;l)$ to $(r,\;s=r+l)$ means that their range will change as follows $$ \left\{ {\matrix{ {1 \le r < \infty } \cr {1 \le l < \infty } \cr } } \right.\quad \Rightarrow \quad \left\{ {\matrix{ {1 \le r < \infty } \cr {r + 1 \le s = r + l < \infty } \cr } } \right.\quad \Rightarrow \quad \left\{ {\matrix{ {1 \le r < s} \cr {2 \le s = r + l < \infty } \cr } } \right. $$ Note that the map $\left( {r,l} \right)\; \leftrightarrow \;\left( {r,s} \right)$ is bi-jective so that we can freely convert $P(r,l)$ to $P(r,s)$ and v.v. $$ P\left( {r,l} \right)\quad \matrix{ {\buildrel {l\, = \,s\, - \,r} \over \longrightarrow } \cr {\mathrel{\mathop{\kern0pt\longleftarrow} \limits_{s\, = \,r + l}} } \cr } \quad P\left( {r,s} \right) $$ and we can convert a sum in $P(r,l)$ to a sum in $P(r,s)$ by applying the conversion above to the summand, and changing the range of summation as specified above.
Analogously can be done for the integral, since the Jacobian of the transformation equals $1$.
We can then write $$ \eqalign{ & E\left( {{r \over {r + l}}} \right) = \left\langle {r/s} \right\rangle = \sum\limits_{l = 1}^\infty {\sum\limits_{r = 1}^\infty {\left( {{r \over {r + l}}} \right)P(r,l)} } = \sum\limits_{r = 1}^\infty {\sum\limits_{l = 1}^\infty {\left( {{r \over {r + l}}} \right)P(r,l)} } = \cr & = \sum\limits_{r = 0}^\infty {\sum\limits_{l = 0}^\infty {\left( {{{r + 1} \over {r + l + 2}}} \right)P(r + 1,l + 1)} } = \cr & = \sum\limits_{s = 2}^\infty {\sum\limits_{r = 1}^{s - 1} {\left( {{r \over s}} \right)P(r,s)} } = \sum\limits_{s = 0}^\infty {\sum\limits_{r = 0}^s {\left( {{{r + 1} \over {s + 2}}} \right)P(r + 1,s + 2)} } \cr} \tag{a.1}$$
b) Upon rescaling the variables dividing by $L$, then for large values of $L$ same as in the precedent post, you can approximately consider $\tilde r=r/L,\; \tilde s =s/L$ to be continuous, i.e. convert the sum in a Riemann sum in $\Delta r/L, \; \Delta s/L$, and approximate with the corresponding integral.
The multiplicative factor is included in the approximation provided for $P(\tilde r, \tilde s)$, which integrates to $1$.
In passing to the integral the small "deviations" in the integration bounds given by the limits in the sum , will be negligible once divided by $L$.
c) Concerning the possibility to evaluate exactly the expected ratio $r/(r+l)$, we are going to have $$ \left\langle {{r \over {r + l}}} \right\rangle = \sum\limits_{r = 0}^\infty {\sum\limits_{l = 0}^\infty {\left( {{{r + 1} \over {r + l + 2}}} \right)P(r + 1,l + 1)} } \tag{c.1}$$
Taking $P(r,l)$ to read as given in the paper $$ P(r,l) = 4pq{{\Gamma (L)} \over {\Gamma (L - 2p)}}{{\Gamma (L + l - 1 - 2p)} \over {\Gamma (L + l - 2q)}} {{\Gamma (L + l + r - 1 - 2q)} \over {\Gamma (L + l + r)}} \tag{c.2}$$ we get $$ \eqalign{ & \left\langle {{r \over {r + l}}} \right\rangle = \sum\limits_{l = 0}^\infty {\sum\limits_{r = 0}^\infty {\left( {{{r + 1} \over {r + l + 2}}} \right)P(r + 1,l + 1)} } = \cr & = 4pq{{\Gamma (L)} \over {\Gamma (L - 2p)}}\sum\limits_{l = 0}^\infty {{{\Gamma (L + l - 2p)} \over {\Gamma (L + l + 1 - 2q)}} \sum\limits_{r = 0}^\infty {\left( {{{r + 1} \over {r + l + 2}}} \right){{\Gamma (L + l + r + 1 - 2q)} \over {\Gamma (L + l + r + 2)}}} } \cr} \tag{c.3}$$
The inner sum becomes $$ \eqalign{ & \sum\limits_{r = 0}^\infty {\left( {{{r + 1} \over {r + l + 2}}} \right){{\Gamma (L + l + r + 1 - 2q)} \over {\Gamma (L + l + r + 2)}}} = \cr & = \left( {{1 \over {l + 2}}} \right){{\Gamma (L + l + 1 - 2q)} \over {\Gamma (L + l + 2)}}\; {}_3F_{\,2} \left( {2,\,2 + l,\,L + l + 1 - 2q\;;\;3 + l,\,L + l + 2\;;\;1} \right) \cr} \tag{c.4}$$ and, although there might be some other ways to put it, I do not see an easy way to express the double sum.
d) Passing to asymptotics, using for $P(r,l)$ the expression provided at the end of eq.(33) $$ P(r,l) \approx 4q\left( {1 - q} \right)L^{\,2q} \left( {L + r} \right)^{\,1 - 4q} \left( {L + r + l} \right)^{\,2q - 3} \tag{d.1}$$ we get $$ \eqalign{ & \left\langle {{r \over {r + l}}} \right\rangle = \sum\limits_{l = 0}^\infty {\sum\limits_{r = 0}^\infty {\left( {{{r + 1} \over {r + l + 2}}} \right)P(r + 1,l + 1)} } = \cr & = 4q\left( {1 - q} \right)L^{\,2q} \sum\limits_{r = 0}^\infty {\left( {L + r + 1} \right)^{\,1 - 4q} \sum\limits_{l = 0}^\infty {\left( {{{r + 1} \over {r + l + 2}}} \right)\left( {L + r + l + 2} \right)^{\,2q - 3} } } = \cr & = 4q\left( {1 - q} \right)L^{\,2q} \sum\limits_{r = 0}^\infty {\left( {L + r + 1} \right)^{\,1 - 4q} \left( {r + 1} \right) \sum\limits_{l = 0}^\infty {{1 \over {\left( {r + l + 2} \right)\left( {L + r + l + 2} \right)^{\,3 - 2q} }}} } \cr} \tag{d.2}$$
From here, and starting from the inner sum, various approaches can be taken and evaluated against the expected error. It is not this the place to conduct such an analysis, so I will limit to hint some possible approach.
For instance, since $3-2q$ ranges from $1$ to $3$, then we can develop the sum $$ \sum\limits_{k = 0}^\infty {{1 \over {\left( {k + a} \right)\left( {k + b} \right)^{\,n} }}} \,\;\left| {\;n = 1,2,3} \right. $$ into partial fractions, solve in terms of the polygamma functions $\psi^{(0)},\,\psi^{(1)},\,\psi^{(2)}$, and interpolate for the actual value of $3-2q$.
Otherwise we can replace the summand with its asymptotic series for $k \to \infty$ $$ \sum\limits_{k = a}^\infty {{1 \over {k\left( {k + b} \right)^{\,c} }}} = \sum\limits_{k = a}^\infty {\sum\limits_{j = 0}^\infty {{{\left( { - 1} \right)^j b^j c^{\,\overline {\,j\,} } } \over {j!k^{\,c + 1 + j} }}} } $$ appropriately truncate it, use the Hurwitz zeta function, ...
e) If, as you requested, we want to work directly on the sum given in c), then we have better and put it under the version $$ \eqalign{ & \left\langle {r/s} \right\rangle = \sum\limits_{s = 0}^\infty {\sum\limits_{r = 0}^s {\left( {{{r + 1} \over {s + 2}}} \right)P(r + 1,s + 2)} } = \cr & = 4pq{{\Gamma (L)} \over {\Gamma (L - 2p)}}\sum\limits_{s = 0}^\infty {{1 \over {s + 2}} {{\Gamma (L+s+1-2q)} \over {\Gamma (L+s+2)}} \sum\limits_{r = 0}^s {\left( {r + 1} \right){{\Gamma (L + s - r - 2 + 2q)} \over {\Gamma (L + s - r + 1 - 2q)}}} } = \cr & = 4\left( {1 - q} \right)q{{\Gamma (L)} \over {\Gamma (L - 2 + 2q)}} \sum\limits_{s = 0}^\infty {{1 \over {s + 2}}\left( {L + s + 2} \right)^{\,\overline {\, - 1 + 2q\,} } \sum\limits_{r = 0}^s {\left( {r + 1} \right)\left( {L + s - r + 1 - 2q} \right)^{\,\overline {\, - 3 + 4q\,} } } } \cr} \tag{e.1}$$ where $x^{\,\overline {\,a\,} }$ and $x^{\,\underline {\,a\,} } $ denote respectively the Rising and Falling factorials.
Now the matter here is that the exponents in $q$ are in general not integer. Therefore
for an "exact" approach I cannot devise other than the approach through the Hypergeometric function already described, for which the second sum would be difficult to express.
going asymptotically, for large $L$ and thus large values of the base argument of the Rising factorial, I can devise only a couple of ways
e.1) asymptotics of Gamma by using $$ \eqalign{ & z^{\,\overline {\,w\,} } \quad \left| \matrix{ \;\left| z \right| \to \infty \hfill \cr \;\left| {\arg (z + w)} \right| < \pi \hfill \cr} \right.\quad \propto \cr & z^{\,w} \left( {1 + {{w\left( {w - 1} \right)} \over {2z}} + {{w\left( {w - 1} \right)\left( {w - 2} \right)\left( {3w - 1} \right)} \over {24z^{\,2} }} + O\left( {w^{\,6} /z^{\,3} } \right)} \right) \cr} \tag{e.2}$$ which, when taking the first term only would give the asymptotics reported by them at the end of eq.(33), but when applied to the $P(r,l)$ that you report here in your post, and not that in the same eq.(33).
So it seems you are right for this: I did not analyze this aspect yet.
e.2) interpolation of solutions with integer exponents
since for $q=0,1/4,1/2,3/4,1$ the factorial can be split in a few terms , it might be possible to absorb the $(r+1)$ factor in front of it by partial fractions and express the inner sum with a few terms in $\psi$, to be then interpolated ...
However everything depends on what is your goal: better approximation ?
f) Upon further investigation, the most promising approach seems that can be derived from these integral representations of the Rising Factorial $$ x^{\,\underline {\,y\,} } \quad \left| \matrix{\;x,y \in \; \mathbb R \hfill \cr \; - 1 < y \hfill \cr} \right. = {{\Gamma (y + 1)} \over {2\,\pi }}\int_{ - \,\pi }^\pi {e^{\, - \,i\,y\,t} \left( {1 + e^{\,i\,t} } \right)^{\,x} dt} \tag{f.1}$$ derived from the representation of the Binomial, and $$ z^{\,\overline {\, - w\,} } \quad \left| \matrix{ \;0 < {\mathop{\rm Re}\nolimits} (z) \hfill \cr \;0 < {\mathop{\rm Re}\nolimits} (w) \hfill \cr} \right.\quad = {1 \over {\Gamma (w)}}\int_{\,0}^{\,1} {t^{\,z - 2} \left( {{{1 - t} \over t}} \right)^{\,w - 1} dt} \tag{f.2}$$ derived from that of the Beta function.
Then, for instance, the inner sum in (e.1) becomes $$ \eqalign{ & \sum\limits_{r = 0}^s {\left( {r + 1} \right)\left( {L + s - r + 1 - 2q} \right)^{\,\overline {\, - \left( {3 - 4q} \right)\,} } } \quad \left| {\,q < 3/4} \right.\quad = \cr & = \sum\limits_{r = 0}^s {\left( {r + 1} \right){1 \over {\Gamma (3 - 4q)}}\int_{\,0}^{\,1} {t^{\,L + s - r + 1 - 2q} \left( {{{1 - t} \over t}} \right)^{\,2 - 4q - 1} {{dt} \over {t^{\,2} }}} } = \cr & = {1 \over {\Gamma (3 - 4q)}}\int_{\,0}^{\,1} {\left( {\sum\limits_{r = 0}^s {\left( {r + 1} \right)t^{\, - r - 2} } } \right) \left( {{{1 - t} \over t}} \right)^{\,2 - 4q - 1} t^{\,L + s + 1 - 2q} dt} = \cr & = {1 \over {\Gamma (3 - 4q)}}\int_{\,0}^{\,1} {{d \over {dt}}\left( {\sum\limits_{r = 0}^{s + 1} {t^{\, - r} } } \right) \left( {{{1 - t} \over t}} \right)^{\,1 - 4q} t^{\,L + s + 1 - 2q} dt} = \cr & = {1 \over {\Gamma (3 - 4q)}}\int_{\,0}^{\,1} {\left( {{{\left( {s + 2} \right)t - 1 - s - t^{\,s + 2} } \over {t^{\,s + 2} \left( {1 - t} \right)^{\,2} }}} \right)\left( {{{1 - t} \over t}} \right)^{\,1 - 4q} t^{\,L + s + 1 - 2q} dt} = \cr & = - {1 \over {\Gamma (3 - 4q)}}\int_{\,0}^{\,1} {{{\left( {\left( {s + 1} \right) - \left( {s + 2} \right)t + t^{\,s + 2} } \right)t^{\,L + 2q - 2} } \over {\left( {1 - t} \right)^{\,1 + 4q} }}dt} \cr} $$ Since the exponent of $(1-t)$ is negative, it seems that we cannot convert the integral to a combination of Beta functions: to be checked. However we can take due advantage of the fact that $0 \le t \le 1$ to develop the integral in series.
Similarly for the outer summation.