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In many tetration/power tower integrals, one sees a general form of the following. Let this new function be notation used to show the connection between the general result and special cases using types of Incomplete Gamma functions.

The goal is to find an integral representation of the general case or a special case.

If this is not possible, then maybe a special case of it has an integral representation. Note there are ways to put the summand into other functions, but this way is simple. Please note that I will use a made up general “T” function to show how each integral below it is a special case of the following:

$$T_{p,q}^{r,s}\left(_{\ \ a,b,c,d}^{A,B,C,D}\right)=\sum_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b,cn+d)}$$

Here is motivation that integral representations are possible. Note that I will use the primitive for simplicity:

$$\int (cx)^{ax^b}dx=\sum_{n=0}^\infty\frac{(-a)^n Q(n+1,-(bn+1)\ln(cx))}{c^{bn+1}(bn+1)^{n+1}}= \frac{1}{-ac^{b-1}}\sum_{n=0}^\infty\frac{ (-ac^bbn-ac^b)^{n+1}Γ(n+1,-bn\ln(cx)-\ln(cx))}{Γ(n+1,0)} = -\frac{1}{ac^{b-1}} T_{-abc^b,-ac^b}^{1,1}\left(_{\ \ 1,1,0,0}^{1,1,-b\ln(cx),-\ln(cx)}\right) $$

$$\int a^{ta^t}dt=t+\frac{1}{\ln(a)}\sum_{n=0}^\infty \frac {(-1)^n Q(n+1,-nt\,\ln(a))}{n^{n+1}}= t-\frac{1}{\ln(a)}\sum_{n=0}^\infty \frac {(-n)^{-n-1} Γ(n+1,-nt\,\ln(a))}{Γ(n+1,0)} =t-\frac{1}{\ln(a)} T_{-1,0}^{-1,-1}\left(_{\ \ 1,1,0,0}^{1,1,-t\,\ln(a),0}\right) $$

$$\int \frac{dx}{xe^x-1}=\sum_{n=0}^\infty \frac{Γ(n+1,-nx)}{n^{n+1}}= \sum_{n=0}^\infty n^{-n-1} Γ(n+1,-nx)= T_{1,0}^{-1,-1}\left(_{0,1,0,0}^{1,1,-x,0}\right) $$

$$\int \text W(\ln(x))dx=\text W(\ln(x))(x-1)+\sum_{n=1}^\infty\frac{(-1)^n Q(n+1,-n\,\text W(\ln(x))}{n^{n+1}}= \text W(\ln(x))(x-1) -\sum_{n=1}^\infty\frac{(-n)^{-n-1}Γ(n+1,-n\,\text W(\ln(x))}{Γ(n+1)} = T_{-1,0}^{-1,-1}\left(_{1,1,0,0}^{1,1, -\,\text W(\ln(x)),0}\right) $$

Miscellaneous sums of interest. Subfactorial: $$\sum_{n=2}^\infty \frac{1}{!n}=e\sum_{n=0}^\infty \frac{1}{Γ(n+2,-1)}=e\,T_{p,q}^{0,0}\left(_{1,2,0,-1}^{0,1,0,0}\right)$$

$$\sum_{n=-\infty}^{-1} Γ(n,n)=\sum_{n=0}^\infty Γ(-n-1,-n-1)= T_{p,q}^{0,0}\left(_{-1,-1,-1,-1}^{\quad 0,1,0,0}\right) $$

I already know about the Abel-Plana formula, but it offers no new insights. There are other theorems that could possibly be used. How can the integral representations for the goal sum be found? If the integral representation is indeed impossible, then what is an integral representation for a special case? This will help us solve similar problems. Please correct me and give me feedback!

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    $\begingroup$ Did you just randomly spotted the $T$ function? Or is the a motivation? $\endgroup$ Sep 9 at 16:00
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    $\begingroup$ @RounakSarkar As said in the bounty and question, it is a function that I made up just to show how each example is a special case of the general case. It is a motivation. Do you know how to possibly put it in an integral form? $\endgroup$ Sep 9 at 20:44
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    $\begingroup$ Nah bro. I am not intelligent enough. But looking at YOUR question history, it seems that you'll find out what you want very quickly. $\endgroup$ Sep 10 at 13:55
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    $\begingroup$ @RounakSarkar What do you mean by “you’ll find out what you want very quickly”? This website is just a hobby. Thanks for the support. $\endgroup$ Sep 10 at 14:48
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    $\begingroup$ @RounakSarkar One last thing, there seems to have been an upvote from you associated with this inverse gamma integral question. I have checked part of the expansion and it works. Can you please check if the rest works? Thanks. $\endgroup$ Sep 11 at 2:30
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This will be a very general solution. Notice we can answer any question like this one by:

$$\int f(x) dx=g(x)=\sum_{n=0}^\infty g_n(x)\implies f(x)=\sum_{n=0}^\infty g’_n(x) =g(x)$$

However, we need to find g(x) in closed form. I could do the general summation in the question by finding a general solution family, but it is more beautiful when we use the special case which can be simplified using a function.

This function will be the Mittag-Leffler function:

$$\mathrm E_{a,b}(y)=\sum_{n=0}^\infty \frac{y^n}{Γ(an+b)}$$

Now let’s solve for $y=y(x)$ by making a simple integral equation. Let’s also make each constant a function of x, so $p=p(x),q=q(x),r=r(x),s=s(x),A=A(x),B=B(x),C=C(x),D=D(x)$

$$\int E_{a,b}(y(x))dx=\sum_{n=0}^\infty \int \frac{y^n(x)}{Γ(an+b)}dx=\sum_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b)}$$

I could take the nth root, but the index must stay with the summation.

After playing around with more generalized y(x), a solution was found, but two of the arguments must be 1 unfortunately unless you can find a better solution. The integration can be found using gamma identities, but I had machine help:

$$\int \mathrm E_{a,b}\left(\ln^t(ux)x^v\right)dx=\sum_{n=0}^\infty \frac{1}{Γ(an+b)}dx\int \ln^{tn}(ux)x^{vn} dx=C+\sum_{n=0}^\infty \frac{x^{n v} (u x)^{-n v} \ln^{n t}(u x) (-(n v + 1) \ln(u x))^{-n t} Γ(n t + 1, -(n v + 1) \ln(u x)) }{Γ(an+b) (n u v + u)}\quad\mathop=^{\ln^{n t}(u x) \ln^{-n t}(u x)=1}_{x^{nv}x^{-nv}=1}\quad C+\sum_{n=0}^\infty \frac{(-1)^{nt} Γ(tn + 1, -(n v + 1) \ln(u x)) }{Γ(an+b) (n v + 1)^{tn+1} u^{vn+1}} = C+\frac1u\sum_{n=0}^\infty \frac{\left(-u^{\frac vt}vn - u^{\frac vt}\right)^{-tn} Γ(tn + 1, -v\ln(u x) n -\ln(u x))) }{Γ(an+b) (n v + 1)} $$

This expansion has a factor of $(nv+1)$ in the denominator which makes it difficult to put into my “T function form”. Here is a better result using the Productlog/W-Lambert function which also uses a bit of software help:

$$\int \mathrm E_{a,b}\left(\mathrm W^{t}(ux)x^v\right)dx=\int\sum_{n=0}^\infty \frac{\mathrm W^{tn}(ux)x^{un}}{Γ(a,b)}dx=C+\sum_{n=0}^\infty \frac{\left[x^{n v} \mathrm W(u x)^{n t} e^{-n v \mathrm W(u x)} (-(n v + 1) \mathrm W(u x))^{-n (t + v)} ((n v + 1) Γ(n (t + v) + 1, -(n v + 1) \mathrm W(u x)) - Γ(n (t + v) + 2, -(n v + 1) \mathrm W(u x)))\right]}{u (n v + 1)^2 Γ(an+b)}$$

If terms can be cancelled out then, the following holds also using Lambert-W function properties:

$$\int \mathrm E_{a,b}\left(\mathrm W^{t}(ux)x^v\right)dx=C+ \sum_{n=0}^\infty \frac{\left[-u^{-vn-1} (-(n v + 1))^{-n (t + v)-1} Γ(n (t + v) + 1, -(n v + 1) \mathrm W(u x)) -u^{-vn-1}(-(n v + 1))^{-n(t+v)-2} Γ(n (t + v) + 2, -(n v + 1) \mathrm W(u x))]\right]}{Γ(an+b)} = C-u^{-\frac{t}{t+v}}\sum_{n=0}^\infty \frac{\left(-u^{\frac{v}{t+v}} n v - u^{\frac{v}{t+v}} \right)^{-n (t + v)-1} Γ(n (t + v) + 1, -n v \mathrm W(u x) -\mathrm W(u x)))}{Γ(an+b)}- u^{\frac{v-t}{t+v}}\sum_{n=0}^\infty \frac{\left(-u^{\frac{v}{t+v}} n v - u^{\frac{v}{t+v}} \right)^{-n (t + v)-1} Γ(n (t + v) + 2, -n v \mathrm W(u x) -\mathrm W(u x)))}{Γ(an+b)} $$

So we mainly found an integral representation for:

$$\sum_{n=0}^\infty \frac{(pn+q)^{rn+\{-1,0,1,2\}}Γ(An+\{1,2\},Bn+C)}{Γ(an+b)}$$

where $\{x_1,x_2,…x_i\}$ represents solved cases of a variable. If you can find a more general solution, then please let me know. Note that the Generalized Exponential Integral function simplifies the problem, but that is not the focus of the problem. Please correct me and give me feedback!

I actually wanted a closed form for this sum and I found an analogous function called the Incomplete Fox-Wright functions where the subscripted variable are constants with respect to $n$. The bolded link has more identities. Also note the Lower Gamma function:

$$\,_pΨ_q^{(\gamma)}\left[\,^{(a_1,A_1,x),(a_2,A_2),…(a_p,A_p)}_{\quad(b_1,B_1),…,(b_p,B_p)}\ t\right]+ \,_pΨ_q^{(\Gamma)}\left[\,^{(a_1,A_1,x),(a_2,A_2),…(a_p,A_p)}_{\quad(b_1,B_1),…,(b_p,B_p)}\ t\right]\mathop=^\text{def}\sum_{n=0}^\infty \frac{\Gamma(a_1+A_1n,x)\prod\limits_{j=2}^pΓ(a_j+A_jn)t^n}{\prod\limits_{j=2}^p \Gamma(b_j+B_jn)n!}+ \sum_{n=0}^\infty \frac{γ(a_1+A_1n,x)\prod\limits_{j=2}^pΓ(a_j+A_jn)t^n}{\prod\limits_{j=2}^p \Gamma(b_j+B_jn)n!} = \sum_{n=0}^\infty \frac{\prod\limits_{j=1}^pΓ(a_j+A_jn)t^n}{\prod\limits_{j=2}^p \Gamma(b_j+B_jn)n!} = \,_pΨ_q\left[\,^{(a_1,A_1,x),(a_2,A_2),…(a_p,A_p)}_{\quad(b_1,B_1),…,(b_p,B_p)}\ t\right] $$

Which is the Fox-Wright function inspired from @Harry Peter’s post and Therefore:

$$\sum_{n=0}^\infty\frac{\Gamma(A+Bn,C)t^n\Gamma(1n+1)}{\Gamma(аn+b)n!}= \sum_{n=0}^\infty\frac{\Gamma(A+Bn,C)t^n}{\Gamma(а+bn)} =\,_2Ψ^{(\Gamma)}_1\left[\,^{(A,B,C),(1,1)}_{\quad(a,b)}\ t\right]$$

which is still not enough to solve general tetration type integrals, but works for some problems. Maybe I will post some later. This function seem like an excuse for any similar sum, but still can be simplified into a few simpler function.

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  • $\begingroup$ I knew that you would find a way to get an answer. Remember my second comment in the question. $\endgroup$ Sep 14 at 3:33
  • $\begingroup$ @RounakSarkar note that “c=d=0” and “B=s=1”, but it should work for most problems as usually “tetration type” integrals have these properties. Thanks. $\endgroup$ Sep 14 at 11:54
  • $\begingroup$ Yep. You are getting closer. $\endgroup$ Sep 14 at 14:29
  • $\begingroup$ Have you made any progress on this function? By the way, did you name the function $T$ because your user name starts with $T$? $\endgroup$ Sep 27 at 6:32
  • $\begingroup$ @RounakSarkar No, I named it for “tetration” becuase of the title. Let me see the progress… $\endgroup$ Sep 27 at 18:27

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