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The increasing price of gasoline is imposing a heavy burden on consumers and industrial consumers. Especially with the recent sharp rise in oil price, the burden on customers in a number of countries has become very serious. Under such circumstances, in recent years, even in the case of vehicles driven by a combustion engine of the gasoline direct injection (GDI) system, in particular, a direct injection engine having high-pressure injection in the combustion chamber, there has been remarkable progress in fuel cost reduction by setting, on a large scale, injection or the like with the view to reducing the emission of CO2 generated by combustion. For example, such technology is disclosed in Patent Literature 1 and Patent Literature 2.
In Patent Literature 1, on the basis of a target fuel injection amount Tf and a value of a variable electric current flowing through a current sensor for a current supervisory/regulating device, the supervisory/regulating device controls the fuel injection amount. Thus, the target fuel injection amount Tf in Patent Literature 1 is the amount of fuel to be injected.
On the other hand, in Patent Literature 2, the supervisory/regulating device calculates a predetermined fuel injection amount and sets a target injection amount from this injection amount. In Patent Literature 2, the predetermined fuel injection amount is expressed as Tf and the target injection amount is expressed as Tg.Q:

Finding an asymptotic expansion of a formula for the gamma function

I recently came across the following formula, defined as
$$\gamma(n+1)=\sum_{k=0}^n\binom{n+1}{k}\frac{\gamma}{k+1}$$
and used to find recurrence relations for the gamma function. I’m not looking for a closed-form expansion, as these are given elsewhere on the interwebs (I’m sure). I’m looking for an asymptotic expansion around $n=\infty$, as there should be a relatively simple way to bound the size of the sum. I have done a little bit of work and failed. I had hoped to find some exponential decay of the sum and get a result on the order of $n^2$ or $n$ but I’m obviously going about this wrong.
Anybody know a good resource for finding such expansions?

A:

The summand is bounded by $0$ and $1$, so the sum itself is bounded by
\sum_{k=0

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