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\textbf{On Euler's differential method for continued fractions}
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\textbf{M. Jafari Shah Belaghi}$^{1}$ and A. Bashirov$^{1}$

$^{1}$Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Turkey (TRNC)
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\textbf{Abstract}

A continued fraction is an expression of the form
\[
 a_0+K_{k=1}^\infty \bigg[ \frac{b_k}{a_k}\bigg] =a_0+\frac{b_1}{a_1+\frac{b_2}{a_2+\frac{b_3}{a_3+\cdots }}},
\]
where $a_0,a_1,a_2,\ldots $ and $b_1,b_2,b_3,\ldots $ are two sequences of real or complex numbers. It is remarkable that rational and irrational numbers can be clearly distinguished by continued fractions. In the 19th century, theory of continued fractions was one of the most popular areas of investigation in mathematics. The great mathematicians such as Karl Jacobi, Oscar Perron, Charles Hermit, Karl Friderich Gauss, Augustin Cauchy, Thomas Stieltjes etc. have contributed to the theory [1].

We study the continued fraction $f(t)=K_{k=1}^\infty \big[ \frac{k+t}{k}\big] $, which depends on the parameter $-1<t<\infty $. This continued fraction was studied by Euler. Using the Euler's differential method, which was not used by mathematicians for a long time, we derive a new formula
\[
f(t)=\frac{\int _0^1(1-x)^{p-t}\frac{d^p}{dx^p}(x^{t+1}e^x)\,dx}{\int _0^1(1-x)^{p-t}\frac{d^p}{dx^p}(x^te^x)\,dx},\ p-1<t\le p+1,
\]
where $p=0,1,2,\ldots \,$. For the integer values $t=p=1,2,\ldots \,$,
\[
f(p)=(p+1)\frac{\sum _{k=0}^{p-1}\frac{a_{p,k}}{p-k+1}}{\sum _{k=0}^{p-1}a_{p,k}},
\]
where
\[
a_{p,k}=\left( \begin{array}{c} p \\ k\end{array}\right) \cdot\frac{1}{(p-k-1)!}.
\]
Previously, it was proved by Euler that $f(0)=(e-1)^{-1}$. Using numerical methods it is found that the function $\sigma (t)=\sqrt{t}-f(t)$ is slowly increasing and $\lim _{t\to \infty}\sigma (t)=0.25$.

\textbf{References}

[1] Khrushchev, S.V., Orthogonal Polynomials and Continued Fractions from Euler's Point of View, Encyclopedia of Mathematics and Its Applications, Vol.~122., Cambridge University Press 2008.


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