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1 Applicable Analysis and Discrete Mathematics available online at Appl. Anal. Discrete Math. x (xxxx, xxx xxx. doi:.98/aadmxxxxxxxx A STUDY OF GENERALIZED SUMMATION THEOREMS FOR THE SERIES F WITH AN APPLICATIONS TO LAPLACE TRANSFORMS OF CONVOLUTION TYPE INTEGRALS INVOLVING KUMMER S FUNCTIONS F Gradimir V. Milovanović, Rakesh K. Parmar, Arun K. Rathie Motivated by recent generalizations of classical theorems for the series F Integral Transform. Spec. Funct. 9(, (, and interesting Laplace transforms of Kummer s confluent hypergeometric functions obtained by Kim et al. Math. Comput. Modelling 55 (, 68 7, first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer s confluent hypergeometric function.. INTRODUCTION AND PRELIMINARIES We begin with the definition of the generalized hypergeometric function with p numerator and q denominator parameters is defined by 7 ( a,...,a p z (a pf q = b,...,b p F q q (b z := (a m (a p m z m (b m m (b q m m!, where the Pochhammer symbol is defined by (c =, (c n = c(c+ (c+n, and b C\Z, =,s :=,,...,s. The series convergesfor all z C if p q. It is divergent for all z when p > q +, unless at least one numerator parameter Mathematics Subect Classification. 33C, 33C5, 33C9. Keywords and Phrases. Bailey s summation theorem, Gauss s second summation theorem, Kummer s summation theorem, Generalized summation theorem, Kummer s confluent hypergeometric function, Laplace transform.

2 G. V. Milovanović, R. K. Parmar, A. K. Rathie is a negative integer in which case ( is a polynomial. Finally, if p = q +, the series converges on the unit circle z = when Re ( b a >. The importance of the generalized hypergeometric function lies in the fact that almost all elementary functions such as exponential, binomial, trigonometric, hyperbolic, logarithmic etc. are special case of this function. From the application point of view, it is very important to represent the hypergeometric function by the well-known gamma function. Thus the well known classical summation theorems such as those of Gauss second summation theorem, Bailey summation theorem and Kummer s summation theorem for the series F which are given below, play an important role in the theory of hypergeometric functions. Gauss s second summation theorem ( F Bailey s summation theorem (3 F a, a b a,b (a+b+ = Γ( Γ( (a+b+ Γ( a+ Γ( b+, = and Kummer s summation theorem (4 F a,b +a b Γ( bγ( b+ Γ( b+ aγ( b a+ = Γ(+ aγ(+a b Γ(+aΓ(+ a b. During 99 96, in a series of three research papers, Lavoie et al. 3, 4, 5 generalized various classical summation theorems such as the Gauss second, Bailey and Kummer ones for the F series, as well as the Watson, Dixon and Whipple ones for the 3 F series. In our present investigation, we are interested in the following generalizations of Gauss s second, Bailey and Kummer summation theorems, respectively, recorded in 5, (5 F a,b (a+b+i+ = Γ( Γ( a+ b+ i+ Γ( a b i+ Γ( a b+ i + { A i (a,b Γ( a+ Γ( b+ i+ +i + B i (a,b Γ( aγ( b+ i i = L i (a,b,

3 Generalized summation theorems for the series F and applications 3 (6 and (7 F F a, a+i b Γ( = Γ(bΓ( a { C i (a,b b i Γ( a+ i+ i Γ( b a+ Γ( b+ a +i D i (a,b + Γ( b aγ( b+ a i = M i (a,b, a,b +a b+i = Γ( Γ( bγ(+a b+i { E i (a,b a Γ( b+ i+ i Γ( a b+ i+γ( a+ i+ +i F i (a,b + Γ( a b+ i+ Γ( a+ i i = N i (a,b. For i =, these formulas respectively give classical Gauss s second summation theorem, Bailey s summation theorem and Kummer s summation theorem, given by (, (3 and (4, respectively (cf. 8. Here, and in what follows, as usual, x denotes the greatest integer less than or equal to the real number x and its modulus is denoted by x. The coefficients which appears in (5, (6 and (7 were determined and listed for i 5, in the papers 5 and. In, Rakha and Rathie6 generalized above mentioned classical theorems in the most genral form for any i Z and established six general summation theorems (two each. For details we refer their original paper 6. The aim of this research paper is two fold. First we provide the explicit expressions for all pairs of coefficients {A i (a,b,b i (a,b, {C i (a,b,d i (a,b and {E i (a,b,f i (a,b in (5, (6 and (7, respectively, thereafter, we demonstrate how one can easily obtain Laplace transforms of convolution type integral by using product theorem for Laplace transforms for a pair of Kummer s confluent hypergeometric function by employing generalized summation formulas (5, (6 and (7. The paper is organized as follows. In Section, we derive explicit expressions for these coefficients for eachinteger i =,±,±,...in the generalizedsummation formulas(5, (6and(7. ByusingproducttheoremforLaplacetransformforapair of Kummer s confluent hypergeometric functions, presented shortly in Section 3, in Section4weemploythe generalizedsummation formulas(5, (6 and (7 inorderto derive new Laplace transforms of convolution type integrals involving F (a;b;x. Some interesting special cases are also included. Finally, some concluding remarks and observations are presented in Section 5.

4 4 G. V. Milovanović, R. K. Parmar, A. K. Rathie. EXPLICIT EXPRESSIONS FOR COEFFICIENTS IN THE GENERALIZED SUMMATION FORMULAS.. Generalized Gauss s second summation theorem Theorem.. For each ν N, the coefficients A i (a,b and B i (a,b in (5 are given by ( ( ( ν b a+ A ν (a,b = A ν (a,b = (ν, = ν ν ( ( ν b+ ( a B ν (a,b = B ν (a,b = + = (ν, ν ( ( ( ν + b+ a+ A (ν+ (a,b = A ν+ (a,b = (ν, + = ν ( ( ν + b ( a B (ν+ (a,b = B ν+ (a,b = (ν. ν = Proof. We start with a recent result by Rakha and Rathie 6, Theorem, (8 where F a,b (a+b+i+ = Γ( S i (a,b = Γ ( (a+b+i+ Γ ( (a b i+ Γ ( b Γ ( (b+ Γ ( (a b+i+ S i(a,b, i r= ( i ( r Γ ( (b+r r Γ ( (a i+r +. Taking even and odd indices r in these sums for i = ν and i = ν +, we have S ν (a,b = and S ν+ (a,b = = = respectively. Since ( ν Γ ( b+ Γ ( (a+ (ν ν = ( ( ν Γ (b++ + Γ ( a (ν ( ( ν + Γ b+ ( ( ν + Γ Γ ( (b++ a (ν + Γ ( (a+ (ν, = Γ(z + = (z Γ(z and Γ(z = (z, Γ(z

5 Generalized summation theorems for the series F and applications 5 the previous sums reduce to S ν (a,b = and Γ( b Γ( (a+ = ( ν Γ( (b+ ν Γ( a S ν+ (a,b = Γ( b Γ( a = = ( ν + ( ( b a+ (ν ν ( b+ ( ν + ( b = ( a (ν ( a (ν ν Γ( (b+ ν ( ( ( ν + b+ a+ Γ( (a+ + ν (ν. ν (9 Now, comparing (8 with (5 for i, i.e., F a,b (a+b+i+ { = Γ( Γ ( (a+b+i+ Γ ( (a b i+ A i (a,b Γ ( (a+ Γ ( b+s i+ Γ ( (a b+i+ B i (a,b + Γ ( a Γ ( b+s, i where s i = i/ i/ (i.e., for even and / for odd i, we obtain the explicit expressions for the coefficients A i (a,b and B i (a,b, for even and odd indices, given in the statement of this theorem. In a similar way, using 6, Theorem, we get the coefficients for negative indices. Corollary.. For each ν N we have B ν+ (b,a = A ν+ (a,b, i.e., the following identity ( holds. ( ( ν + b = = ( a (ν = ν ( ( ( ν + b+ a+ + (ν ν Proof. Note first that the interchange of parameters a and b does not change the value of the hypergeometric function in (9, and put, for i = ν +, K ν+ (a,b = Γ( ( Γ (a+b+ν + Γ ( (a b ν Γ (. (a b+ν +

6 6 G. V. Milovanović, R. K. Parmar, A. K. Rathie This constant can be written in the form ( ( a+b a+b πγ ν K ν+ (a,b = ( ( a b a b ν ν+ ν = ( ( a+b a+b πγ ν ( a b ν, a b = from which we conclude that K ν+ (b,a = K ν+ (a,b. Now, interchanging a and b in (9 for i = ν +, we get that B ν+ (b,a = A ν+ (a,b. According to Theorem. we conclude that the identity ( is valid for each ν N. Remark.. Following Theorem. the coefficients in (5 are: A (a,b =, A (a,b =, A (a,b = (a+b, A 3 (a,b = (3a+b, A 4(a,b = ( a +6ab+b 4a 4b+3, 4 A 5 (a,b = ( 5a +ab+b a 6b+8, 4 A 6 (a,b = 8 (a+b ( a +4ab 8a+b 8b+5, A 7 (a,b = 8 (3a+b 4(a+b (a b +6 and + 4 (a+b (a b 4, A 8 (a,b = 6 (a b (a b 6 + (a+b (a+b (a b ,... B (a,b =, B (a,b =, B (a,b =, B 3 (a,b = (a+3b, B 4 (a,b = (a+b, B 5 (a,b = ( a +ab+5b 6a b+8, 4 B 6 (a,b = (a b 4(a+b 9, B 7 (a,b = 8 (a+3b 4(a+b (a b +6 4 (a+b (a b 4, B 8 (a,b = (a+b (a b (a+b 7,....

7 Generalized summation theorems for the series F and applications 7.. Generalized Bailey s summation theorem Using a similar procedure as in the previous subsection, as well as Theorems 5 and 6 from 6, we can prove the following generalization of Bailey s summation theorem: Theorem.. For each ν N, the coefficients C i (a,b and D i (a,b in (6 are given by C ν (a,b = ( ( ( ν b a b+a = = (ν, ν ν ( ( ( ν b a+ b+a+ D ν (a,b = + C ν+ (a,b = D ν+ (a,b = C ν (a,b = = ( ( ( ν + b a b+a = ( ( ( ν + b a+ b+a+ + ( ( ( ν b a b+a = = +, ν ν ( ( ( ν b a+ b+a+ D ν (a,b = + C (ν+ (a,b = D (ν+ (a,b = ( ( ( ν + b a b+a = = (ν, ν (ν +, ν +, ν ( ( ( ν + b a+ b+a+ + (ν +, ν +, ν +. ν Remark.. For i 7, the coefficients C i (a,b and D i (a,b are: C (a,b =, C (a,b =, C (a,b = b, C 3 (a,b = a b+3, C 4 (a,b = a +5a+ ( b 6b+6, C 5 (a,b = a +a(b 3 4b +b, C 6 (a,b = (b 4 ( 3a +a+4b 3b+3, C 7 (a,b = a 3 +a (4b 6+a ( 4b 68b+8 8b 3 +9b 88b+;

8 8 G. V. Milovanović, R. K. Parmar, A. K. Rathie C (a,b =, C (a,b = b, C 3 (a,b = b a, C 4 (a,b = a(a+3+b(b+, C 5 (a,b = a a(b+7+4b(b+, C 6 (a,b = (b+ ( 3a 5a+4b(b+4, C 7 (a,b = a 3 a (4b+ a ( b +b+3 +8b ( b +6b+8 ; D (a,b =, D (a,b =, D (a,b =, D 3 (a,b = a+b 7, D 4 (a,b = 4(b 3, D 5 (a,b = a +a(b +4b 34b+6, D 6 (a,b = ( a 7a 4b +3b 54, D 7 (a,b = a 3 +a (3 4b+a ( 4b 4b 7 +8b 3 4b +576b 76; D (a,b =, D (a,b =, D 3 (a,b = a+b+, D 4 (a,b = 4(b+, D 5 (a,b = a +a(b +4(b+(b+3, D 6 (a,b = 8(b+(b+3 a(a+5, D 7 (a,b = a 3 a (4b+9+a(4(b b 58+8(b+(b+3(b Generalized Kummer s summation theorem Also, in similar way as before and using Theorems 3 and 4 from 6, we can prove the following generalization of Kummer s summation theorem: Theorem.3. For each ν N, the coefficients E i (a,b and F i (a,b in (6 are given by E ν (a,b = = ( ( ( ν a b+ a+ +ν (ν, ν ν ( ( ν a b ( a F ν (a,b = +ν + + (ν, ν E ν+ (a,b = F ν+ (a,b = E ν (a,b = F ν (a,b = = = = = ( ( ν + a b = ( a +ν + (ν, ν ( ( ( ν + a b+ a+ +ν + (ν, + ν ( ( ( ν a b+ a+ ν (ν, ν ν ( ( ν a b ( a (ν + (ν, ν

9 Generalized summation theorems for the series F and applications 9 E (ν+ (a,b = F (ν+ (a,b = ( ( ν + a b = = ( a ν (ν, ν ( ( ( ν + a b+ a+ ν (ν. + ν Remark.3. For i 7, the coefficients E i (a,b and F i (a,b are: E (a,b =, E (a,b =, E (a,b = a b+, E 3 (a,b = a+3b 5, E 4 (a,b = a 4a(b +b 3b+,, E 5 (a,b = 4a +a(5b 3 5b +5b 3, E 6 (a,b = 4a 3 a (b 3+a ( 9b 5b+74 b 3 +6b b+6, E 7 (a,b = 8a 3 +4a (7b 5 4a ( 7b 49b+88 +7b 3 7b +45b 3; E (a,b =, E (a,b = a b, E 3 (a,b = a 3b 4, E 4 (a,b = a 4a(b++b +5b+6, E 5 (a,b = 4a a(5b++5b +5b+3, E 6 (a,b = 4a 3 a (b+3+a ( 9b +57b+9 b 3 b 47b 6, E 7 (a,b = 8a 3 4a (7b+4+4a ( 7b +49b+88 7b 3 77b 94b 384; F (a,b =, F (a,b =, F (a,b =, F 3 (a,b = a b+, F 4 (a,b = 4(a b+, F 5 (a,b = 4a +a(4 6b+b 3b+, F 6 (a,b = 8a +6a(b 3 6b +34b 5, F 7 (a,b = 8a 3 +a (68 b+4a ( 3b 9b+3 b 3 +6b b+6; F (a,b =, F (a,b =, F 3 (a,b = a b, F 4 (a,b = 4(a b, F 5 (a,b = 4a a(3b+8+b +7b+, F 6 (a,b = 8a 6a(b+3+6b +38b+64, F 7 (a,b = 8a 3 4a (5b+8+4a ( 3b +3b+46 b 3 5b 74b. 3. LAPLACE TRANSFORM OF PRODUCT OF F (a;b;x The Laplace transform (cf. of the function f(t is defined, as usual, by L{f(t = e st f(tdt.

10 G. V. Milovanović, R. K. Parmar, A. K. Rathie Here, we recall the following general product theorem for Laplace transform 9, p. 43, Eq. 3..8: { t g (tg (t = e st f (τf (t τdτ dt, which further applied to a pair of generalized hypergeometric functions to obtain the formula 9, p. 43, Eq. 3..9: { t (a e st τ µ (t τ ν (a AF B kτ A (b F B (b k (t τ dτ dt = Γ(µΓ(νs µ ν A+F B (a,µ (b k (a A s +F,ν B (b for A < B, A < B, Re(µ >, Re(ν >, Re(s > or for A = B, A = B, Re(µ >, Re(ν >, Re(s > Re(k, Re(s > Re(k. If A = B = = A = B, we get known formula 9, p. 43, Eq. 3..3: ( { t e st τ µ (t τ ν a F c kτ = Γ(µΓ(νs µ ν F a,µ c k s, a k F c (t τ dτ dt k F a,ν;c k s s for Re(c >, Re(c >, Re(s > Re(k, Re(s > Re(k, s > k and s > k. In the next section we present various new Laplace type integrals by using product theorem for Laplace transform for a pair of Kummer s confluent hypergeometric functions by employing generalized Gauss s second summation theorem, Bailey s summation theorem and Kummer s summation theorem, (5, (6 and (7, with coefficients given by Theorems.,. and.3, respectively. 4. NEW LAPLACE TRANSFORMS OF CONVOLUTION TYPE INTEGRALS INVOLVING F (a;b;x Theorem 4.. For Re(b >, Re(b > and Re(s >, the following general result ( { t e st τ b (t τ b F F a (a+b+i+ a (a +b + + = Γ(bΓ(b s b+b L i (a,bl (a,b, τs (t τs dτ dt

11 Generalized summation theorems for the series F and applications hold, where L i (a,b is given in (5. Proof. The proof of this theorem is quite straight forward. In order to prove this result, setting k = s, k = s, µ = b, ν = b, c = (a + b + i + and c = (a +b + + for i, Z in (, we have (3 { t e st τ b (t τ b F F = Γ(bΓ(b s b+b F a a (a+b+i+ (a +b + + a,b (a+b+i+ τs (t τs dτ F dt a,b (a +b + +. We, now observe that the two F appearing on the right-hand side of (3 can be evaluated with the help of generalized Gauss s second summation theorem (5, which yields at once the desired formula (. The following results can also be proven in a similar lines by applying appropriate summation theorems (5, (6 and (7. Theorem 4.. For Re( a+i >, Re( a + > (i, Z and Re(s >, the following general result holds true: { t e st τ a+i (t τ a + a F b where M i (a,b is given in (6. a τs F b (t τs dτ dt = Γ( a+iγ( a + s i++ a a M i (a,bm (a,b, Theorem 4.3. For Re(b >, Re(b > and Re(s >, the following general result holds true: (4 { t e st τ b (t τ b F F a +a b + a +a b+i (t τs dτ τs dt where N i (a,b is given in (7. = Γ(bΓ(b s b+b N i (a,bn (a,b,

12 G. V. Milovanović, R. K. Parmar, A. K. Rathie Theorem 4.4. For Re(b >, Re( a + > (i, Z and Re(s >, the following general result holds true: { t e st τ b (t τ a + a F (a+b+i+ a F b (t τs dτ dt = Γ(bΓ( a + s + a +b L i (a,bm (a,b, where L i (a,b and M i (a,b are given in (5 and (6, respectively. τs Theorem 4.5. For Re(b >, Re(b > and Re(s >, the following general result holds true: (5 { t e st τ b (t τ b F F a +a b + = Γ(bΓ(b s b+b L i (a,bn (a,b, a (a+b+i+ τs (t τs dτ where L i (a,b and N i (a,b are given in (5 and (7, respectively. Theorem 4.6. For Re( a + i > (i Z Re(b > and Re(s >, the following general result holds true: dt (6 { t e st τ a+i (t τ b a F b F τs a +a b + (t τs dt = Γ( a+iγ(b s i+ a+b M i (a,bn (a,b, where M i (a,b and N i (a,b are given in (6 and (7, respectively. 5. SPECIAL CASES For i =, Theorems 4. to 4.6 yield the following interesting results for classical ones asserted in the following corollaries:

13 Generalized summation theorems for the series F and applications 3 Corollary 5.. For Re(b >, Re(b > and Re(s >, the following result holds true: { t e st τ b (t τ b F F = Γ(bΓ(b s b+b a (a+b+ a (a +b + τs (t τs dτ Γ( Γ( (a+b+ Γ( a+ Γ( b+ Γ( Γ( (a +b + Γ( a + Γ( b +. dt Corollary 5.. For Re( a >, Re( a > and Re(s >, the following result holds true: { t e st τ a a (t τ a F a b τs F b (t τs dτ dt = Γ( aγ( a s a a Γ( bγ( b+ Γ( b+ aγ( b a+ Γ( b Γ( b + Γ( b + a Γ( b a +. Corollary 5.3. For Re(b >, Re(b > and Re(s >, the following result holds true: { t e st τ b (t τ b F = Γ(bΓ(b s b+b F a +a b τs a (t τs +a b dτ Γ(+ aγ(+a b Γ(+aΓ(+ a b Γ(+ a Γ(+a b Γ(+a Γ(+ a b. dt Corollary 5.4. For Re(b >, Re( a > and Re(s >, the following result holds true: { t e st τ b a (t τ a F (a+b+ a τs F b (t τs dτ dt = Γ(bΓ( a s a +b Γ( Γ( (a+b+ Γ( a+ Γ( b+ Γ(+ a Γ(+a b Γ(+a Γ(+ a b.

14 4 G. V. Milovanović, R. K. Parmar, A. K. Rathie Corollary 5.5. For Re(b >, Re(b > and Re(s >, the following result holds true: { t e st τ b (t τ b F F = Γ(bΓ(b s b+b a (a+b+ τs a (t τs +a b dτ Γ( Γ( (a+b+ Γ( a+ Γ( b+ Γ(+ a Γ(+a b Γ(+a Γ(+ a b. dt Corollary 5.6. For Re( a >, Re(b > and Re(s >, the following result holds true: { t e st τ a (t τ b a F b F = Γ( aγ(b s a+b τs a (t τs +a b dτ dt Γ( Γ( (a+b+ Γ(+ a Γ(+a b Γ( a+ Γ( b+ Γ(+a Γ(+ a b. Similarly, for i Z other results can also be obtained. 6. CONCLUDING REMARKS AND OBSERVATIONS It is interesting to mention here that the following three results which are closely related to (4, (5 and (6 can also be obtained on similar lines. These are given here without proof. Theorem 6.. For Re(a >, Re(b > and Re(s >, the following general result holds true: { t e st τ a (t τ b F F a +a b + where N i (a,b is given in (7. (τ ts dτ b +a b+i τs dt = Γ(aΓ(b s a+b N i (a,bn (a,b,

15 Generalized summation theorems for the series F and applications 5 Theorem 6.. For Re(a >, Re(b > and Re(s >, the following general result holds true: { t e st τ a (t τ b F F a +a b + (τ ts dτ b (a+b+i+ τs dt = Γ(aΓ(b s a+b L i (a,bn (a,b, where L i (a,b and N i (a,b are given in (5 and (7, respectively. Theorem 6.3. For Re( a+i > (i Z! { t e st τ a+i (t τ a a F b τs F = Γ( a+iγ(a s i+ a+a M i (a,bn (a,b, b +a b + where M i (a,b and N i (a,b are given in (6 and (7, respectively. (τ ts dt Acknowledgments. The first author was supported in part by the Serbian Academy of Sciences and Arts (No. Φ-96 and by the Serbian Ministry of Education, Science and Technological Development (No. #OI 745. REFERENCES. M. Abramowitz, I. A. Stegun (Eds.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 97.. Y. S. Kim, A. K. Rathie, D. Cviović: New Laplace Transforms of Kummer s confluent hypergeometric functions. Math. Comput. Modelling, 55 (, J. L. Lavoie, F. Grondin, A. K. Rathie: Generalizations of Watson s theorem on the sum of a 3F. Indian J. Math. 34 (99, J. L. Lavoie, F. Grondin, A. K. Rathie, K. Arora: Generalizations of Dixon s theorem on the sum of a 3F. Math. Comp. 6 (994, J. L. Lavoie, F. Grondin, A. K. Rathie: Generalization of Whipple s theorem on the sum of a 3F. J. Comput. Appl. Math. 7 (996, M. A. Rakha, A. K. Rathie: Generalizations of classical summation theorems for the series F and 3F with applications. Integral Transform. Spec. Funct. 9 (, (, E. D. Rainville: Special Functions. The Macmillan Co., New York 96; Reprinted by Chelsea Publishing Company, Bronx, New York, L. J. Slater: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge, 966.

16 6 G. V. Milovanović, R. K. Parmar, A. K. Rathie 9. L.J. Slater: Confluent Hypergeometric Functions, Cambridge University Press, New York, 96.. I. N. Sneddon: The Use of the Integral Transforms. Tata McGraw-Hill, New Delhi, 979. Serbian Academy of Sciences and Arts, Beograd, Serbia University of Niš, Faculty of Sciences and Mathematics, Niš, Serbia Department of Mathematics, Government College of Engineering and Technology Bikaner-3344, Raasthan State, India Department of Mathematics, School of Physical Sciences Central University of Kerala, Kasaragod-6736, India