# Applications of Lehmer’s Infinite Series Involving Reciprocals of the Central Binomial Coefficients

Authors: B. R. Srivatsa Kumar; Adem Kılıçman; Arjun K. Rathie
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric functionFq+1qz for q=2,3,4,5. This is achieved by means of separating the generalized hypergeometric function Fq+1qz (q=2,3,4,5) into even and odd components together with the use of several known infinite series involving reciprocals of the central binomial coefficients obtained earlier by Lehmer.


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## Body

## 1. Lehmer’s Series Involving Central Binomial Coefficients

The binomial coefficients are defined by(1)nk=n!k!n−k!,n≥k,0,n<k,for nonnegative integers n and k. The central binomial coefficients are defined by
(2)2nn=2n!n!2n=0,1,2,⋯.It is well-known that the binomial and reciprocal of binomial coefficients play an important role in many areas of mathematics (including number theory, probability, and statistics). Actually, the sums containing the central binomial coefficients and reciprocals of the central binomial coefficients have been studied for a long time. A large number of very interesting results can be seen in the research papers by Lehmer [1], Mansour [2], Pla [3], Rockett [4], Sprugnoli [5, 6], Sury [7], Sury et al. [8], Trif [9], Wheelon [10], and Zhao and Wang [11]. Many facts about the central binomial coefficients and the reciprocals of the central binomial coefficients can be found in the book of Koshy [12]. Gould [13] has collected numerous identities involving central binomial coefficients. Riordan [14] is also a good reference. However, in our present investigation, we are interested in a very interesting paper due to Lehmer [1] in which he studied the following two types of the series, viz,
(3)∑n=0∞an2nn,∑n=0∞an2nn,where an are of very simple functions of n and deduced several interesting series involving the central binomial coefficients and reciprocals of the central binomial coefficients, with Golden ratio φ=5+1/2 as follows:
(4)∑n=1∞2nnn4n=ln4,(5)∑n=1∞−1n−12nnn4n=2ln2+12,(6)∑n=1∞1n2nn=π39,(7)∑n=1∞−1n−1n2nn=25lnφ,(8)∑n=1∞1n22nn=π218,(9)∑n=1∞−1n−1n22nn=2lnφ2,(10)∑n=1∞12nn=131+2π39,(11)∑n=1∞−1n−12nn=151+455lnφ,(12)∑n=1∞n2nn=231+3π9,(13)∑n=1∞−1n−1n2nn=212515+25lnφ,(14)∑n=1∞n22nn=28153π+54,(15)∑n=1∞−1n−1n22nn=41255−5lnφ,(16)∑n=1∞n32nn=2243373π+405,(17)∑n=1∞−1nn32nn=26255+145lnφ,(18)∑n=1∞2nn2nn=π2,(19)∑n=1∞−1n−12nn2nn=13ln2+3,(20)∑n=1∞2n2nn=π2+1,(21)∑n=1∞−1n−12n2nn=13+39ln2+3,(22)∑n=1∞n2n2nn=π+3,(23)∑n=1∞−1n−1n2n2nn=13,(24)∑n=1∞n22n2nn=7π2+11,(25)∑n=1∞−1n−1n22n2nn=191−33ln2+3,(26)∑n=1∞n32n2nn=35π2+55,(27)∑n=1∞−1n−1n32n2nn=−18115+3ln2+3,We conclude this section by remarking that in the next section, the results (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), AND (27) will be written in terms of a generalized hypergeometric function.


## 2. Generalized Hypergeometric Function

The generalized hypergeometric functionpFqz with p numerator and q denominator parameters is defined by [15]
(28)Fpqa1,a2,⋯,apb1,b2,⋯,bq;z=∑n=0∞a1na2n⋯apnb1nb2n⋯bqnznn!,where an is the well-known Pochhammer’s symbol defined by
(29)an=aa+1⋯a+n−1,n∈ℕ,1,n=0.In terms of gamma function, we have(30)an=Γa+nΓa.Here, as usual,p and q are nonnegative integers, and the parameters aj1≤j≤p and bj1≤j≤q can have arbitrary complex values with zero or negative integer values of bj excluded. The generalized hypergeometric function Fpqz converges for z<∞,p≤q, z<1p=q+1, and z=1p=q+1andRes>0, where s is the parametric excess defined by
(31)s=∑j=1qbj−∑j=1paj.It is interesting to mention here that the generalized hypergeometric function occurs in many theoretical and practical applications such as mathematics, theoretical physics, engineering, and statistics.For more details about this function, we refer the standard texts [16–21]. Furthermore, it is not difficult to see that the results (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), and (27) given in the previous section can be written in terms of the following generalized hypergeometric function that will be required in our present investigations. These are
(32)F321,1,322,2;1=4ln2,(33)F321,1,322,2;−1=4ln2+12,(34)F211,1,32;14=23π9,(35)F211,1,32;−14=45lnφ,(36)F321,1,132,2;14=π29,(37)F321,1,132,2;−14=4lnφ2,(38)F211,2,32;14=231+23π9,(39)F211,2,32;−14=251+455lnφ,(40)F212,2,32;14=4273π+9,(41)F212,2,32;−14=412515+25lnφ,(42)F322,2,21,32;14=48153π+54,(43)F322,2,21,32;−14=81255−5lnφ,(44)F432,2,2,21,1,32;14=4243373π+405,(45)F432,2,2,21,1,32;−14=−46255+145lnφ,(46)F211,132;12=π2,(47)F211,132;−12=13ln2+3,(48)F211,232;12=π2+1,(49)F211,232;−12=13+39ln2+3,(50)F212,232;12=π+3,(51)F212,232;−12=13,(52)F322,2,21,32;12=7π2+11,(53)F322,2,21,32;−12=19−327ln2+3,(54)F432,2,2,21,1,32;12=35π2+55,(55)F432,2,2,21,1,32;−12=−18115+3ln2+3.Also, it is well-known that the process of resolving a generalized hypergeometric functionFpqz into even and odd components can lead to two new results. We shall employ this procedure combined with the results (32), (33), (34), (35), (36), (37), (38), (39), (40), (41), (42), (43), (44), (45), (46), (47), (48), (49), (50), (51), (52), (53), (54), and (55) to obtain several new and interesting closed form evaluations of the series Fp+1qz for q=1,2,3,4, and 5 with the arguments 1/16, 1/4, 1/2, and 1.


## 3. Several Closed-Form Evaluations

In this section, we shall establish the following twenty-four new closed-form evaluations for the generalized hypergeometric function.(56)F3212,34,5432,32;1=2ln2+1,(57)F431,1,54,7432,2,2;1=1632ln2−ln2+1,(58)F3212,1,134,54;116=3π9+25lnφ,(59)F321,1,3254,74;116=2π3−65lnφ,(60)F4312,12,1,134,54,32;116=12π29+4lnφ2,(61)F431,1,1,3254,74,2;116=6π29−4lnφ2,(62)F321,1,3234,54;116=815+23π27+4525lnφ,(63)F321,32,254,74;116=6115+3π27−2525lnφ,(64)F431,1,32,3212,34,54;116=23π27+6875+45125lnφ,(65)F3232,2,254,74;116=3π9+1625−65125lnφ,(66)F541,1,32,32,3212,12,34,54;116=103π81+11275−45125lnφ,(67)F4332,2,2,21,54,74;116=53π54+2225+35125lnφ,(68)F651,1,32,32,32,3212,12,12,34,54;116=743π243+1244375−285625lnφ,(69)F5432,2,2,2,21,1,54,74;116=373π324+157125+2151250lnφ,(70)F321,1,3234,54;14=12π2+39ln2+3+43,(71)F321,32,254,74;14=34π2−39ln2+3+23,(72)F3212,1,134,54;14=12π2+13ln2+3,(73)F321,1,3254,74;14=32π2−13ln2+3,(74)F431,1,32,3212,34,54;144=π2+53,(75)F3232,2,254,74;14=3π8+1,(76)F541,1,32,32,3212,12,34,54;14=127π2+1009−327ln2+3,(77)F4332,2,2,21,54,74;14=3167π2+989+327ln2+3,(78)F651,1,32,32,32,3212,12,12,34,54;14=35π4+74027−3162ln2+3,(79)F5432,2,2,2,21,1,54,74;14=116105π4+7459+354ln2+3,Proof.
In order to establish the results (56), (57), (58), (59), (60), (61), (62), (63), (64), (65), (66), (67), (68), (69), (70), (71), (72), (73), (74), (75), (76), (77), (78), and (79), we shall employ the procedure of resolving a generalized hypergeometric function Fpqz into even and odd components. This decomposition is facilitated by use of the identities:
(80)a2n=22na2na2+12n,a2n+1=a22na2+12na2+1n,then for the generalized hypergeometric function
(81)q+1Fqa1,a2,⋯,aq+1b1,b2,⋯,bq;z.It is not difficult to see the following relations:
(82)q+1Fqa1,a2,⋯,aq+1b1,b2,⋯,bq;z+q+1Fqa1,a2,⋯,aq+1b1,b2,⋯,bq;−z=22q+2F2q+1a12,a12+12,⋯,aq+12,aq+12+1212,b12,b12+12,⋯,bq2,bq2+12;z2,(83)q+1Fqa1,a2,⋯,aq+1b1,b2,⋯,bq;z−q+1Fqa1,a2,⋯,aq+1b1,b2,⋯,bq;−z=2za1a2⋯aq+1b1b2⋯bq2q+2F2q+1a12+12,a12+1,⋯,aq+12+12,aq+12+132,b12+12,b12+1,⋯,bq+12+12,bq2+1;z2,provided for z=1, the convergence condition for the series q+1Fq (1) should be
(84)∑j=1qbj−∑j=1q+1αj>0.
Therefore, for the derivation of the results (56) and (57), we substitute the results (32) and (33) into the results (82) and (83) with q=2 and setting a1=a2=1,a3=3/2, b1=b2=2,z=1, and we immediately obtain the results ((56)) and ((57)), respectively. The remaining results (58), (59), (60), (61), (62), (63), (64), (65), (66), (67), (68), (69), (70), (71), (72), (73), (74), (75), (76), (77), (78), and (79) can be proven on similar lines, so we left this as an exercise to the interested reader.
We conclude this section by mentioning that the results (56), (57), (58), (59), (60), (61), (62), (63), (64), (65), (66), (67), (68), (69), (70), (71), (72), (73), (74), (75), (76), (77), (78), and (79) established in this paper have been verified by using MAPLE.


## 4. Concluding Remark

In this paper, several new closed-form evaluations of the generalized hypergeometric functionsq+1Fqz for q=1,2,3,4,5 with arguments 1/16, 1/4, 1/2, and 1 have been established. This is achieved by means of separating the generalized hypergeometric function q+1Fqz into even and odd components together with the use of the several known results of interesting series involving central binomial coefficients obtained earlier by Lehmer. We believe that the results established in this paper have not appeared in the literature and represent a definite contribution to the theory of generalized hypergeometric function. It is hoped that the results could be of potential use in the area of mathematics, statistics, and mathematical physics.



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