Arithmetic Means for a Class of Functions and the Modified Bessel Functions of the First Kind

In the paper, by virtue of the residue theorem in the theory of complex functions, the authors establish several identities between arithmetic means for a class of functions and the modified Bessel functions of the first kind, present several identities between arithmetic means for a class of functions and infinite series, and find several series expressions for the modified Bessel functions of the first kind.


Preliminaries
Recall from ( [1], Chapter 6) and ( [2], Chapter 5) that the classical Euler gamma function can be defined by (w) > 0 or by and that the logarithmic derivative is called the psi or digamma function.One can also find these notions in the papers [3,4], the handbooks [1,2], and closely related references therein.
In most calculus texts, the quantity 1 b − a b a f (x) d x is called the arithmetic mean for f (x) on the interval [a, b].This is justified by noting that the approximating Riemann sums are the arithmetic means of values of f (x) at points distributed across [a, b].See ( [5], p. 368).

A Definition and a Lemma
In this paper, one of our main tools is the residue theorem in the theory of complex functions.
be its Laurent expansion about w = w 0 .Then the residue of f (w) at w = w 0 is defined to be the coefficient c −1 and is denoted by Lemma 1 (Residue theorem ( [21], p. 112) and ( [2], p. 19)).Let f be analytic in the domain D except for the isolated singularities w 1 , w 2 , . . ., w n .If γ is a simple closed contour in D and does not pass through any of the points w k for 1 ≤ k ≤ n, then

Several Identities and Series for Arithmetic Means
We now state and prove the first identity for the arithmetic mean Q(α, β; 1, 1) and the modified Bessel function of the first kind I 0 (w) as follows.
Theorem 1.For α, β > 0, we have Proof.Let i = √ −1 and w = e iφ .Then Accordingly, by direct calculation and Lemma 1, we have Straightforward computation yields where any empty sum is taken to be zero.Therefore, by Definition 1, we obtain Substituting this equation into (4) results in the identity (2).The proof of Theorem 1 is thus complete.
We now state and prove the second and third identities between the arithmetic means Q(α, β; 1, 2) and Q(α, β; 2, 1) and two infinite series as follows.
In the second and fourth sums above, all terms are not constant and dependent of the variable w.
The first and third sums above can be rewritten as Consequently, the constant term in the Laurent expansion of the function α where we used the formula The Equation ( 5) is thus proved.
As did in the above or in the proof of Theorem 1, we can also obtain w2(m+n+1) .
In the second and third sums above, all terms are not constant and dependent of the variable w.
The first and fourth sums above can be formulated as Consequently, the constant term in the Laurent expansion of the function α where we used the formula (7).In conclusion, it follows that The Equation ( 6) is thus proved.The proof of Theorem 2 is complete.
We now state and prove an identity between the arithmetic mean Q(α, β; k, k) and an infinite series as follows.
Theorem 3.For α, β > 0 and k ∈ N, we have Proof.Employing the transforms in (3) and Lemma 1 reveals Easy computation gives It is easy to see that, in the above sum ∑ ∞ j=0 ∑ 2j+1 q=0 , there is no any constant term independent of w.On the other hand, because where any empty sum is taken to be zero and ( r s ) = 0 for s > r ≥ 0 and r, s ∈ N, we have It is not difficult to see that, in the sum , there is no any constant term independent of w.In a word, the constant term independent of w in the Laurent expansion of the function (ln α) q (ln β) 2j−q i (2j−q)k (2j)!4 kj kj ∑ r=0 (−1) kj−r kq r k(2j − q) kj − r .
Consequently, by Definition 1, we deduce Res Substituting the identity ( 7) into (10) and simplifying leads to Substituting this equation into (9) leads to (8).The proof of Theorem 3 is thus complete.
We now state and prove an identity between the arithmetic mean Q(α, β; k, ) and an infinite series as follows.
Theorem 4. For α, β > 0 and k, ∈ N, we have Proof.Utilizing the transforms in (3) and Lemma 1 results in Direct computation gives Therefore, by Definition 1, we deduce Res Substituting this result into (12) leads to the identity (11).The proof of Theorem 4 is thus complete.

Two New Series Expressions for I 0
Combining the Equation ( 8) in Theorem 3 with the identities (2) and ( 1) in sequence, we can find two series expressions for I 0 as follows.Theorem 5.For x ∈ R, we have Further letting α → 1 and β = e x for x ∈ R reduces to (13).The proof of Theorem 5 is complete.Theorem 6.For x ∈ R, we have which can be rewritten as (14).The proof of Theorem 6 is complete.

)Proof.
When taking k = 1, the right hand side of the Equation (8) in Theorem 3 becomes sum is taken to be zero and ( r s ) = 0 for s > r ≥ 0 and r, s ∈ N.This means thatI 0 (ln α) 2 + (ln β) 2 =

)Proof.
When taking k = 2, the right hand side of the Equation (8) in Theorem 3 becomes letting β = e ±2x results in e ±x I 0 (x) =