Evaluation of the relationship between the prealbumin/fibrinogen ratio and diabetic nephropathy in patients with type 2 diabetes mellitus

Burcin Meryem Atak Tel; Ramiz Tel; Tuba Duman; Satilmis Bilgin; Hamza Kaya; Halil Bardak; Gulali Aktas; Burcin Meryem Atak Tel; Ramiz Tel; Tuba Duman; Satilmis Bilgin; Hamza Kaya; Halil Bardak; Gulali Aktas

doi:10.3934/medsci.2024008

AIMS Medical Science

2024, Volume 11, Issue 2: 90-98. doi: 10.3934/medsci.2024008

Previous Article Next Article

Research article Topical Sections

Evaluation of the relationship between the prealbumin/fibrinogen ratio and diabetic nephropathy in patients with type 2 diabetes mellitus

1.
Department of Internal Medicine, Abant Izzet Baysal University Hospital, Golkoy, Bolu, Turkey
2.
Department of Emergency Medicine, Izzet Baysal State Hospital, Bolu, Turkey

Received: 27 December 2023 Revised: 17 April 2024 Accepted: 11 May 2024 Published: 17 May 2024

Introduction

Our aim was to compare the prealbumin/fibrinogen ratio (PFR) of diabetic patient populations with or without diabetic nephropathy.

Materials and methods

People with type 2 diabetes who attended the internal medicine outpatient clinic were enrolled in the study. Two groups were formed according to the proteinuria of the patients: Diabetic nephropathy and non-nephropathy group. Diabetic nephropathy was calculated using the mathematical formula of spot urine albumin/spot urine creatinine x100. Patients with proteinuria above 200 mg/g were considered to have nephropathy. PFR was simply calculated by dividing prealbumin by fibrinogen.

Results

A total of 152 patients who attended our outpatient clinic were enrolled in the study. There were 68 patients in the diabetic nephropathy group and 84 in the non-nephropathy group. The prealbumin/fibrinogen ratios (PFR) were significantly lower in the nephropathic group [0.061 (0.02–0.16)] than the non-nephropathic group [0.0779 (0.01–0.75)] (p = 0.002).

Conclusions

We suggest that decreased levels of PFR can indicate diabetic nephropathy in subjects with type 2 diabetes mellitus.

Keywords:

Citation: Burcin Meryem Atak Tel, Ramiz Tel, Tuba Duman, Satilmis Bilgin, Hamza Kaya, Halil Bardak, Gulali Aktas. Evaluation of the relationship between the prealbumin/fibrinogen ratio and diabetic nephropathy in patients with type 2 diabetes mellitus[J]. AIMS Medical Science, 2024, 11(2): 90-98. doi: 10.3934/medsci.2024008

Related Papers:

[1]	Qiuxia Hu, Bilal Khan, Serkan Araci, Mehmet Acikgoz . New double-sum expansions for certain Mock theta functions. AIMS Mathematics, 2022, 7(9): 17225-17235. doi: 10.3934/math.2022948
[2]	Suxia Wang, Tiehong Zhao . New refinements of Becker-Stark inequality. AIMS Mathematics, 2024, 9(7): 19677-19691. doi: 10.3934/math.2024960
[3]	Zhenhua Su, Zikai Tang, Hanyuan Deng . Higher-order Randić index and isomorphism of double starlike trees. AIMS Mathematics, 2023, 8(12): 31186-31197. doi: 10.3934/math.20231596
[4]	Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri . Spectral tau solution of the linearized time-fractional KdV-Type equations. AIMS Mathematics, 2022, 7(8): 15138-15158. doi: 10.3934/math.2022830
[5]	Tie-Hong Zhao, Zai-Yin He, Yu-Ming Chu . On some refinements for inequalities involving zero-balanced hypergeometric function. AIMS Mathematics, 2020, 5(6): 6479-6495. doi: 10.3934/math.2020418
[6]	Mustafa Inc, Mamun Miah, Akher Chowdhury, Shahadat Ali, Hadi Rezazadeh, Mehmet Ali Akinlar, Yu-Ming Chu . New exact solutions for the Kaup-Kupershmidt equation. AIMS Mathematics, 2020, 5(6): 6726-6738. doi: 10.3934/math.2020432
[7]	Aslıhan ILIKKAN CEYLAN, Canan HAZAR GÜLEÇ . A new double series space derived by factorable matrix and four-dimensional matrix transformations. AIMS Mathematics, 2024, 9(11): 30922-30938. doi: 10.3934/math.20241492
[8]	Waleed Mohamed Abd-Elhameed, Abdullah F. Abu Sunayh, Mohammed H. Alharbi, Ahmed Gamal Atta . Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation. AIMS Mathematics, 2024, 9(12): 34567-34587. doi: 10.3934/math.20241646
[9]	Ling Zhu . New inequalities of Wilker's type for hyperbolic functions. AIMS Mathematics, 2020, 5(1): 376-384. doi: 10.3934/math.2020025
[10]	Bai-Ni Guo, Dongkyu Lim, Feng Qi . Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions. AIMS Mathematics, 2021, 6(7): 7494-7517. doi: 10.3934/math.2021438

Abstract

Introduction

Our aim was to compare the prealbumin/fibrinogen ratio (PFR) of diabetic patient populations with or without diabetic nephropathy.

Materials and methods

Results

Conclusions

We suggest that decreased levels of PFR can indicate diabetic nephropathy in subjects with type 2 diabetes mellitus.

1. Introduction

In ^[18] Ramanujan showed a total of 17 series for $1/\pi$ but he did not indicate how he arrived at these series. The Borwein brothers ^[5] gave rigorous proofs of Ramanujan's series for the first time and also obtained many new series for $1/\pi.$ Till now, many new Ramanujan's-type series for $1/\pi$ have been published, (see, for example, ^[4,6,8]). Chu ^[7], Liu ^[15,16] and Wei et al. ^[21,22] gave many $\pi$ -formula with free parameters by means of gamma functions and hypergeometric series. Guillera ^[10] proved a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$ . Moreover, Guillera and Zudilin ^[11] outlined an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/\pi$ . Recently, $q$ -analogues of Ramanujan-type series for $1/\pi$ have caught the interests of many authors (see, for example, ^{[9,12,13,14,20,21]}).

Although various definitions for gamma functions are used in the literature, we adopt the following definition [23, p.76]

$\frac 1{\Gamma(z)} = ze^{\gamma z}\prod\limits_{n = 1}^\infty\left(1+\frac zn\right)e^{-\frac zn}$

where $\gamma$ is the Euler constant defined as

$\gamma = \lim\limits_{n\rightarrow \infty}\left(1+ \frac 12+\cdots+ \frac 1n-\log n\right).$

It is easy to verify that $\Gamma(1) = 1, \Gamma({ \frac 12}) = \sqrt\pi$ and $\Gamma({z+1}) = z\Gamma(z).$ It follows that for every positive integer $n$ , $\Gamma(n) = (n-1)!.$

For any complex $\alpha$ , we define the general rising shifted factorial by

$\begin{align} (z)_{{\alpha}} = \Gamma({z+{\alpha}})/\Gamma(z). \end{align}$

(1.1)

Obviously, $(z)_0 = 1.$ For every positive integer $n$ , we have

$(z)_n = \Gamma({z+n})/\Gamma(z) = z(z+1)\cdots(z+n-1)$

and

$(z)_{-n} = \Gamma({z-n})/\Gamma(z) = \frac 1{(z-1)(z-2)\ldots(z-n)}.$

For convenience, we use the following compact notations

$(a_1,a_2,\ldots,a_m)_n = (a_1)_n(a_2)_n\ldots(a_m)_n$

and

$(a)_{(n_1,n_2,\ldots,n_m)} = (a)_{n_1}(a)_{n_2}\ldots(a)_{n_m}.$

Following ^[1,3], the hypergeometric series is defined by

$_{{r+1}}F_{s}\left[\begin{array}{cccc}a_0,&a_1,&\ldots,&a_r\\&b_1,&\ldots,&b_s\end{array};z\right] = \sum\limits_{k = 0}^\infty \frac {(a_0, a_1,\ldots,a_r)_k}{(b_1,\ldots,b_s)_k} \frac {z^k}{k!},$

where $a_i, b_j(i = 0, 1, \ldots, r, j = 1, 2, \ldots, s)$ are complex numbers such that no zero factors appear in the denominators of the summand on the right hand side.

We let $F_{q:s; v}^{p:r; u}\ ({p, q, r, s, u, v\in \Bbb N_0 = \{0, 1, 2, \ldots\}})$ denote a general (Kampé de Fériet's) double hypergeometric function defined by (see ^[2,19])

$\begin{eqnarray*} &&F_{q:s;v}^{p:r;u}\left[\begin{array}{ccc}{\alpha}_1,\ldots,{\alpha}_p:&a_1,\ldots,a_r;&c_1,\ldots,c_u;\\ \beta_1,\ldots,\beta_q:&b_1,\ldots,b_s;&d_1,\ldots,d_v;\end{array}x,y\right]\\&& = \sum\limits_{m,n = 0}^\infty \frac {({\alpha}_1,\ldots,{\alpha}_p)_{m+n} (a_1,\ldots,a_r)_m(c_1,\ldots,c_u)_n}{(\beta_1,\ldots,\beta_q)_{m+n} (b_1,\ldots,b_s)_m(d_1,\ldots,d_v)_n} \frac {x^m}{m!} \frac {y^n}{n!}, \end{eqnarray*}$

where, for convergence of the double hypergeometric series,

$p+r\leq q+s+1{\quad}\text{and}{\quad} p+u\leq q+v+1,$

with equality only when $|x|$ and $|y|$ are appropriately constrained (see, for details, [19,Eq 1.3(29),p.27]).

There exist numerous identities for such series. For example, we have

Theorem 1.1 (See [,(30)] ) If $Re(e-d) > 0$ and $Re(d+e-a-b-c) > 0,$ then

$F_{1:1;1}^{0:3;3}\left[\begin{array}{ccc}-:&a,b,c;&d-a,d-b,d-c;\\d:&e;&d+e-a-b-c;\end{array}1,1\right] = \frac {\Gamma({e})\Gamma({e+d-a-b-c})\Gamma({e-d})}{\Gamma({e-a})\Gamma({e-b})\Gamma({e-c})}.$

In ^[15], Liu used the general rising shifted factorial and the Gauss summation formula to prove the following four-parameter series expansions formula, which implies infinitely many Ramanujan type series for $1/\pi$ and $\pi$ .

Theorem 1.2 For any complex ${\alpha}$ and $Re(c-a-b) > 0,$ we have

$\sum\limits_{n = 0}^\infty \frac {({\alpha})_{a+n}(1-{\alpha})_{b+n}}{n!\Gamma({c+n+1})} = \frac {({\alpha})_a(1-{\alpha})_b\Gamma({c-a-b})} {({\alpha})_{c-b}(1-{\alpha})_{c-a}}\cdot \frac {\sin \pi{\alpha}}\pi.$

Motivated by Liu's work, in this paper we derive the following result from Theorem 1.1 which enables us to give many double series expansions for $1/\pi$ and $\ \pi$ . To the best of our knowledge, most of the results in this paper have not previously appeared.

Theorem 1.3 If $d\in \Bbb N_0, Re(e-d+{\sigma}-{\delta}) > 0$ and $Re(d+e-a-b-c+{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}) > 0,$ then

$\begin{align*} &\sum\limits_{m,n = 0}^\infty \frac {({\alpha})_{a+m}(\beta)_{b+m}({\gamma})_{c+m}({\delta}-{\alpha})_{d-a+n}({\delta}-\beta)_{d-b+n}({\delta}-{\gamma})_{d-c+n}} {m!n!({\delta}+d)_{m+n}({\sigma})_{e+m}({\delta}+{\sigma}-{\alpha}-\beta-{\gamma})_{d+e-a-b-c+n}}\\ & = \frac {({\alpha})_a(\beta)_b({\gamma})_c({\delta}-{\alpha})_{d-a}({\delta}-\beta)_{d-b}({\delta}-{\gamma})_{d-c}({\sigma}-{\delta})_{e-d}} {({\sigma}-{\alpha})_{e-a}({\sigma}-\beta)_{e-b}({\sigma}-{\gamma})_{e-c}} \cdot \frac {\Gamma({\sigma})\Gamma({{\sigma}-{\delta}})\Gamma({{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}})}{\Gamma({{\sigma}-{\alpha}})\Gamma({{\sigma}-\beta})\Gamma({{\sigma}-{\gamma}})}. \end{align*}$

Several examples of such formulae are

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_m^3( \frac 12)_n^2}{m!n!(m+n)!(m+1)!(2n+1)} = \frac 4{\pi},$

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 32)_n^3}{m!n!(m+n)!(n+3)!( \frac 12)_{m+1}} = \pi,$

and

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^2( \frac 13)_n^3}{m!n!(n+1)!(2-3m)(- \frac 13)_{m+n}} = \frac {\sqrt3\pi}3.$

The remainder of the paper is organized as follows. In section 2 we give the proof of Theorem 1.3. Sections 3 and 4 are devoted to the double series expansions for $1/\pi$ and $\pi$ , respectively.

2. Proof of Theorem 1.3

First of all, by making use of (1.1), Theorem 1.3 can be restated as follows:

$\begin{equation} \begin{split} &\sum\limits_{m,n = 0}^\infty \frac {\Gamma({a+m})\Gamma({b+m})\Gamma({c+m})\Gamma({d-a+n})\Gamma({d-b+n})\Gamma({d-c+n})} {m!n!\Gamma({d+m+n})\Gamma({e+m})\Gamma({d+e-a-b-c+n})}\\ & = \frac {\Gamma(a)\Gamma(b)\Gamma(c)\Gamma({d-a})\Gamma({d-b})\Gamma({d-c})\Gamma({e-d})}{\Gamma(d)\Gamma({e-a})\Gamma({e-b})\Gamma({e-c})}. \end{split} \end{equation}$

(2.1)

From (1.1) it is easy to see that

$\begin{align*} &\Gamma({a+{\alpha}+m}) = ({\alpha})_{a+m}\Gamma({{\alpha}}), \ \Gamma({b+\beta+m}) = (\beta)_{b+m}\Gamma({\beta}),\ \Gamma({c+{\gamma}+m}) = ({\gamma})_{c+m}\Gamma({{\gamma}}),\\ &\Gamma({d-a+{\delta}-{\alpha}+n}) = ({\delta}-{\alpha})_{d-a+n}\Gamma({{\delta}-{\alpha}}),\ \Gamma({d-b+{\delta}-\beta+n}) = ({\delta}-\beta)_{d-b+n}\Gamma({{\delta}-\beta}),\\ &\Gamma({d-c+{\delta}-{\gamma}+n}) = ({\delta}-{\gamma})_{d-c+n}\Gamma({{\delta}-{\gamma}}),\ \Gamma({d+{\delta}+m+n}) = ({\delta})_{d+m+n}\Gamma({{\delta}})\\ &\Gamma({e+m+{\sigma}}) = ({\sigma})_{e+m}\Gamma({\sigma}),\ \Gamma({a+{\alpha}}) = ({\alpha})_a\Gamma({\alpha}),\ \Gamma({b+\beta}) = (\beta)_b\Gamma(\beta),\ \Gamma({c+{\gamma}}) = ({\gamma})_c\Gamma({\gamma}),\\ &\Gamma({d-a+{\delta}-{\alpha}}) = ({\delta}-{\alpha})_{d-a}\Gamma({{\delta}-{\alpha}}),\ \Gamma({d-b+{\delta}-\beta}) = ({\delta}-\beta)_{d-b}\Gamma({{\delta}-\beta}),\\ &\Gamma({d-c+{\delta}-{\gamma}}) = ({\delta}-{\gamma})_{d-c}\Gamma({{\delta}-{\gamma}}),\ \Gamma({e-d+{\sigma}-{\delta}}) = ({\sigma}-{\delta})_{e-d}\Gamma({{\sigma}-{\delta}}),\\ &\Gamma({d+{\delta}}) = ({\delta})_d\Gamma({\delta}),{\quad} \Gamma({e-a+{\sigma}-{\alpha}}) = ({\sigma}-{\alpha})_{e-a}\Gamma({{\sigma}-{\alpha}}),\\ &\Gamma({e-b+{\sigma}-\beta}) = ({\sigma}-\beta)_{e-b}\Gamma({{\sigma}-\beta}),\ \Gamma({e-c+{\sigma}-{\gamma}}) = ({\sigma}-{\gamma})_{e-c}\Gamma({{\sigma}-{\gamma}}),\\ &\Gamma({d+e-a-b-c+{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}})\\& = ({\delta}+{\sigma}-{\alpha}-\beta-{\gamma})_{d+e-a-b-c}\Gamma({{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}}). \end{align*}$

and we realize that $({\delta})_{d+m+n} = ({\delta})_d({\delta}+d)_{m+n}$ when $d\in \Bbb N_0.$ Replacing $(a, b, c, d, e)$ by $(a+{\alpha}, b+\beta, c+{\gamma}, d+{\delta}, e+{\sigma})$ in (2.1) and substituting above identities into the resulting equation, we get the desired result.

3. Double series expansions for $1/\pi$

In this section we will use Theorem 1.3 to prove the following double series expansion formula for $1/\pi$ .

Theorem 3.1 If $d\in \Bbb N_0, Re(e-d+1) > 0$ and $Re(d+e-a-b-c+ \frac 32) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(a+m,b+m,c+m,d-a+n,d-b+n,d-c+n)}} {m!n!(d+1)_{m+n}(2)_{e+m}( \frac 32)_{d+e-a-b-c+n}} = \frac {( \frac 12)_{(a,b,c,d-a,d-b,d-c)}(1)_{e-d}} {( \frac 32)_{(e-a,e-b,e-c)}}\cdot \frac 4{\pi}.$

Proof. Let $({\alpha}, \beta, {\gamma}, {\delta}, {\sigma}) = ( \frac 12, \frac 12, \frac 12, 1, 2)$ in Theorem 1.3. We find that

$\begin{align} \sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(a+m,b+m,c+m,d-a+n,d-b+n,d-c+n)}} {m!n!(d+1)_{m+n}(2)_{e+m}( \frac 32)_{d+e-a-b-c+n}} = \frac {( \frac 12)_{(a,b,c,d-a,d-b,d-c)}(1)_{e-d}} {( \frac 32)_{(e-a,e-b,e-c)}}\cdot \frac {\Gamma(2)\Gamma(1)\Gamma({ \frac 32})}{\Gamma^3( \frac 32)}. \end{align}$

(3.1)

Substituting $\Gamma({ \frac 32}) = \frac {\sqrt{\pi}}2$ into (3.1) we obtain the result immediately. Putting $(a, b, c) = (0, 0, 0)$ in Theorem 3.1 we get the following general double summation formula for $1/\pi$ with two free parameters.

Corollary 3.2 If $d\in \Bbb N_0, Re(e-d+1) > 0$ and $Re(d+e+ \frac 32) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(m,d+n)}^3} {m!n!(d+1)_{m+n}(2)_{e+m}( \frac 32)_{d+e+n}} = \frac {4( \frac 12)_d^3(1)_{e-d}}{\pi( \frac 32)_e^3}.$

Setting $d = 0$ and $e = k\in\Bbb N_0$ in Corallary 3.2 we have the following result.

Proposition 3.3 Let $k$ be a nonnegative integer. Then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(m,n)}^3}{m!n!(m+n)!(m+k+1)!( \frac 32+k)_n} = \frac {4k!}{\pi( \frac 32)_k^2}.$

Example 3.1 ( $k = 0$ in Proposition 3.3).

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_m^3( \frac 12)_n^2}{m!n!(m+n)!(m+1)!(2n+1)} = \frac 4{\pi}.$

If $d = e = k\in\Bbb N_0$ in Corollary 3.2 we achieve

Proposition 3.4 Let $k$ be a nonnegative integer. Then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(m,n+k)}^3} {m!n!(k+1)_{m+n}(m+k+1)!( \frac 32)_{n+2k}} = \frac 4{\pi(2k+1)^3}.$

If we put $k = 0$ into Proposition 3.4, then we can also get Example 3.1.

4. Double series expansions for $\pi$

In this section we will prove the following theorem, which allows us to derive infinitely double series expansions for $\pi$ .

Theorem 4.1 If $d\in \Bbb N_0, Re(e-d-{\sigma}+1) > 0$ and $Re(d+e-a-b-c+2) > 0,$ then

$\begin{align*} &\sum\limits_{m,n = 0}^\infty \frac {({\sigma}-1)_{(a+m,b+m,c+m)}({\sigma})_{(d-a+n,d-b+n,d-c+n)}} {m!n!(2{\sigma}+d-1)_{m+n}({\sigma})_{e+m}(2)_{d+e-a-b-c+n}}\\ & = \frac {({\sigma}-1)_{(a,b,c)}({\sigma})_{(d-a,d-b,d-c)}(1-{\sigma})_{e-d}}{(1)_{(e-a,e-b,e-c)}} \cdot \frac {\pi}{\sin{{\sigma}\pi}}. \end{align*}$

Proof. Let $({\alpha}, \beta, {\gamma}, {\delta}) = ({\sigma}-1, {\sigma}-1, {\sigma}-1, 2{\sigma}-1)$ in Theorem 1.3. We obtain that

$\begin{align*} &\sum\limits_{m,n = 0}^\infty \frac {({\sigma}-1)_{(a+m,b+m,c+m)}({\sigma})_{(d-a+n,d-b+n,d-c+n)}} {m!n!(2{\sigma}+d-1)_{m+n}({\sigma})_{e+m}(2)_{d+e-a-b-c+n}}\\ & = \frac {({\sigma}-1)_{(a,b,c)}({\sigma})_{(d-a,d-b,d-c)}(1-{\sigma})_{e-d}} {(1)_{(e-a,e-b,e-c)}} \cdot \frac {\Gamma({\sigma})\Gamma({1-{\sigma}})\Gamma({2})}{\Gamma^3(1)}. \end{align*}$

(4.1)

Combining $\Gamma({{\sigma}})\Gamma({1-{\sigma}}) = \frac \pi{\sin{\sigma}\pi}$ with (4.1) we get the desired result immediately. Putting $a = b = c = 0$ in Theorem 4.1 we obtain the following equation.

Corollary 4.2 If $d\in \Bbb N_0, Re(e-d-{\sigma}+1) > 0$ and $Re(d+e+2) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {({\sigma}-1)_m^3({\sigma})_{d+n}^3} {m!n!(2{\sigma}+d-1)_{m+n}({\sigma})_{e+m}(2)_{d+e+n}} = \frac {({\sigma})_d^3(1-{\sigma})_{e-d}}{(1)_e^3} \cdot \frac {\pi}{\sin{{\sigma}\pi}}.$

4.1. $\sigma=\frac{1}{2}$ in Corollary 4.2

Letting $\sigma=\frac{1}{2}$ in Corollary 4.2, we get the following proposition.

Proposition 4.3 If $d\in \Bbb N_0, Re(e-d+ \frac 12) > 0$ and $Re(d+e+2) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 12)_{d+n}^3}{m!n!(d)_{m+n}( \frac 12)_{e+m} (2)_{d+e+n}} = \frac {( \frac 12)_d^3( \frac 12)_{e-d}}{(1)_{e}^3} \pi.$

When we set $d = 1$ and $e = k\in\Bbb N = \{1, 2, 3\ldots\}$ in Proposition 4.3 we obtain

Proposition 4.4 If $k$ is a positive integer, then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 32)_n^3}{m!n!(m+n)!(n+k+2)!( \frac 12)_{m+k}} = \frac {\pi( \frac 12)_{k-1}}{(k!)^3} .$

Example 4.1 ( $k = 1$ in Proposition 4.4).

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 32)_n^3}{m!n!(m+n)!(n+3)!( \frac 12)_{m+1}} = \pi.$

4.2. $\sigma=\frac{1}{3}$ in Corollary 4.2

Putting $\sigma=\frac{1}{3}$ in Corollary 4.2, we get the following proposition.

Proposition 4.5 If $d\in \Bbb N_0, Re(e-d+ \frac 23) > 0$ and $Re(d+e+2) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 13)_{d+n}^3}{m!n!(d- \frac 13)_{m+n}( \frac 13)_{e+m} (2)_{d+e+n}} = \frac {2\sqrt3\pi( \frac 13)_d^3( \frac 23)_{e-d}}{3(1)_{e}^3}.$

When we set $d = 0$ and $e = k\in\Bbb N_0$ in Proposition 4.5 we obtain

Proposition 4.6 If $k$ is a nonnegative integer, then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 13)_n^3}{m!n!(- \frac 13)_{m+n}( \frac 13)_{m+k}(n+k+1)!} = \frac {2\sqrt3\pi( \frac 23)_k}{3k!^3}.$

Example 4.2 ( $k = 0$ in Proposition 4.6).

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^2( \frac 13)_n^3}{m!n!(n+1)!(2-3m)(- \frac 13)_{m+n}} = \frac {\sqrt3\pi}3.$

Setting $d = e = k\in\Bbb N_0$ in Proposition 4.5, we get

Proposition 4.7 If $k$ is a nonnegative integer, then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 13+k)_n^3}{m!n!(n+2k+1)!(k- \frac 13)_{m+n}( \frac 13)_{m+k} } = \frac {2\sqrt3\pi}{3k!^3}.$

Therefore, Example 4.2 can also be deduced by fixing $k = 0$ in the above equation.

Example 4.3 ( $k = 1$ in Proposition 4.7).

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 43)_n^3}{m!n!(n+3)!( \frac 23)_{m+n} ( \frac 43)_m} = \frac {2\sqrt3\pi}9.$

5. Conclusions

Double series expansions for $1/\pi$ and $\pi$ with several free parameters are established and many interesting formulas are obtained. A point that should be stressed is that there is an important connection between the summation formulas for double hypergeometric functions and double series expansions for the powers of $\pi$ .

Acknowledgements

The author was partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (grant 19KJB110006).

Conflicts of interest

The author declares that there is no conflict of interest in this paper.

Author contributions

Burcin Meryem Atak Tel: study design, data curation, data interpretation, statistical analyses, critical review and writing; Ramiz Tel: study design, statistical analyses, writing; Tuba Duman: formal analysis; Satilmis Bilgin: formal analysis; Hamza Kaya: data curation, writing; Halil Bardak: data curation, data interpretation; Gulali Aktas: study design, formal analysis, critical review and writing. All authors have read and approved the final version of the manuscript for publication.

Data availability

Data will be shared by corresponding author on reasonable request.

Ethics approval of research and informed consent

This work has been approved by Abant Izzet Baysal University ethics committee (approval number: 2022/205). All subjects have consent to participate to the study.

Conflict of interest

The authors declare no conflicts of interest.

References

[1]	Lovic D, Piperidou A, Zografou I, et al. (2020) The growing epidemic of ddiabetes mellitus. Curr Vasc Pharmacol 18: 104-109. https://doi.org/10.2174/1570161117666190405165911
[2]	Arnous MM, Al Saidan AA, Al Dalbhi S, et al. (2022) Association of atrial fibrillation with diabetic nephropathy: A meta-analysis. J Family Med Prim Care 11: 3880-3884. https://doi.org/10.4103/jfmpc.jfmpc_577_21
[3]	Kim YG, Han KD, Roh SY, et al. (2023) Being underweight is associated with increased risk of sudden cardiac death in people with diabetes mellitus. J Clin Med 12: 1045. https://doi.org/10.3390/jcm12031045
[4]	Bailes BK (2002) Diabetes mellitus and its chronic complications. AORN J 76: 266-27. https://doi.org/10.1016/s0001-2092(06)61065-x
[5]	Braunwald E (2019) Diabetes, heart failure, and renal dysfunction: The vicious circles. Prog Cardiovasc Dis 62: 298-302. https://doi.org/10.1016/j.pcad.2019.07.003
[6]	Targher G, Bertolini L, Zoppini G, et al. (2005) Increased plasma markers of inflammation and endothelial dysfunction and their association with microvascular complications in Type 1 diabetic patients without clinically manifest macroangiopathy. Diabet Med 22: 999-1004. https://doi.org/10.1111/j.1464-5491.2005.01562.x
[7]	Spijkerman AM, Gall MA, Tarnow L, et al. (2007) Endothelial dysfunction and low-grade inflammation and the progression of retinopathy in Type 2 diabetes. Diabet Med 24: 969-76. https://doi.org/10.1111/j.1464-5491.2007.02217.x
[8]	Mansoor G, Tahir M, Maqbool T, et al. (2022) Increased expression of circulating stress markers, inflammatory cytokines and decreased antioxidant level in diabetic nephropathy. Medicina (Kaunas) 58: 1604. https://doi.org/10.3390/medicina58111604
[9]	Yue W, Liu Y, Ding W, et al. (2015) The predictive value of the prealbumin-to-fibrinogen ratio in patients with acute pancreatitis. Int J Clin Pract 69: 1121-1128. https://doi.org/10.1111/ijcp.12682
[10]	Hrnciarikova D, Juraskova B, Hyspler R, et al. (2007) A changed view of serum prealbumin in the elderly: prealbumin values influenced by concomitant inflammation. Biomed Pap Med Fac Univ Palacky Olomouc Czech Repub 151: 273-276. https://doi.org/10.5507/bp.2007.046
[11]	Sun DW, An L, Lv GY (2020) Albumin-fibrinogen ratio and fibrinogen-prealbumin ratio as promising prognostic markers for cancers: an updated meta-analysis. World J Surg Oncol 18: 9. https://doi.org/10.1186/s12957-020-1786-2
[12]	Omiya K, Sato H, Sato T, et al. (2021) Albumin and fibrinogen kinetics in sepsis: a prospective observational study. Crit Care 25: 436. https://doi.org/10.1186/s13054-021-03860-7
[13]	Chen C, Liu L, Luo J (2022) Identification of the molecular mechanism and candidate markers for diabetic nephropathy. Ann Transl Med 10: 1248. https://doi.org/10.21037/atm-22-5128
[14]	Hou S, Jin C, Yang M, et al. (2022) Prognostic Value of Hematologic Prealbumin/Fibrinogen Ratio in Patients with Glioma. World Neurosurg 160: e442-e453. https://doi.org/10.1016/j.wneu.2022.01.048
[15]	Bilgin S, Kurtkulagi O, Atak Tel BM, et al. (2021) Does C-reactive protein to serum Albumin Ratio correlate with diabEtic nephropathy in patients with type 2 diabetes MEllitus? The CARE TIME study. Prim Care Diabetes 15: 1071-1074. https://doi.org/10.1016/j.pcd.2021.08.015
[16]	Kocak MZ, Aktas G, Duman TT, et al. (2020) Monocyte lymphocyte ratio as a predictor of diabetic kidney injury in type 2 diabetes mellitus; The MADKID study. J Diabetes Metab Disord 19: 997-1002. https://doi.org/10.1007/s40200-020-00595-0
[17]	Wang J, Xi H, Zhang K, et al. (2020) Circulating C-reactive protein to prealbumin ratio and prealbumin to fibrinogen ratio are two promising inflammatory markers associated with disease activity in rheumatoid arthritis. Clin Lab 66. https://doi.org/10.7754/Clin.Lab.2019.190833
[18]	Zhang HL, Zhang XM, Mao XJ, et al. (2012) Altered cerebrospinal fluid index of prealbumin, fibrinogen, and haptoglobin in patients with Guillain-Barre syndrome and chronic inflammatory demyelinating polyneuropathy. Acta Neurol Scand 125: 129-135. https://doi.org/10.1111/j.1600-0404.2011.01511.x
[19]	Zang S, Shi L, Zhao J, et al. (2020) Prealbumin to fibrinogen ratio is closely associated with diabetic peripheral neuropathy. Endocr Connect 9: 858-863. https://doi.org/10.1530/EC-20-0316
[20]	Rim TH, Byun IH, Kim HS, et al. (2013) Factors associated with diabetic retinopathy and nephropathy screening in Korea: the third and fourth Korea national health and nutrition examination survey (KNHANES III and IV). J Korean Med Sci 28: 814-820. https://doi.org/10.3346/jkms.2013.28.6.814
[21]	Hess K (2015) The vulnerable blood. Coagulation and clot structure in diabetes mellitus. Hamostaseologie 35: 25-33. https://doi.org/10.5482/HAMO-14-09-0039
[22]	Dayer MR, Mard-Soltani M, Dayer MS, et al. (2014) Causality relationships between coagulation factors in type 2 diabetes mellitus: path analysis approach. Med J Islam Repub Iran 28: 59.
[23]	Zhou C, Zhang Y, Yang S, et al. (2022) Associations between visceral adiposity index and incident nephropathy outcomes in diabetic patients: Insights from the ACCORD trial. Diabetes Metab Res Rev 39: e3602. http://doi.org/10.1002/dmrr.3602.e3602
[24]	Aktas G (2023) Association between the prognostic nutritional index and chronic microvascular complications in patients with type 2 diabetes mellitus. J Clin Med 12: 5952. https://doi.org/10.3390/jcm12185952
[25]	Aktas G, Yilmaz S, Kantarci DB, et al. (2023) Is serum uric acid-to-HDL cholesterol ratio elevation associated with diabetic kidney injury?. Postgrad Med 135: 519-523. https://doi.org/10.1080/00325481.2023.2214058

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Medical Science

0.4

Metrics

Article views(1129) PDF downloads(39) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(2) / Tables(1)

AIMS Medical Science

Evaluation of the relationship between the prealbumin/fibrinogen ratio and diabetic nephropathy in patients with type 2 diabetes mellitus

Related Papers:

Abstract

1. Introduction

2. Proof of Theorem 1.3

3. Double series expansions for $1/\pi$

4. Double series expansions for $\pi$

4.1. $\sigma=\frac{1}{2}$ in Corollary 4.2

4.2. $\sigma=\frac{1}{3}$ in Corollary 4.2

5. Conclusions

Acknowledgements

Conflicts of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Medical Science

Evaluation of the relationship between the prealbumin/fibrinogen ratio and diabetic nephropathy in patients with type 2 diabetes mellitus

Related Papers:

Abstract

1. Introduction

2. Proof of Theorem 1.3

3. Double series expansions for 1/π 1/\pi

4. Double series expansions for π \pi

4.1. σ=12 \sigma=\frac{1}{2} in Corollary 4.2

4.2. σ=13 \sigma=\frac{1}{3} in Corollary 4.2

5. Conclusions

Acknowledgements

Conflicts of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

3. Double series expansions for $1/\pi$

4. Double series expansions for $\pi$

4.1. $\sigma=\frac{1}{2}$ in Corollary 4.2

4.2. $\sigma=\frac{1}{3}$ in Corollary 4.2