correspondence to the following recycled solution where and Hereby, as long as the reference Schr¨ odinger equation (13) is in point, the corre- sponding solutions hold true. However, a mapping to the original target problem defined through equations (4)-(6) would suggest that and with 6 φ ( q ) = (cosh µq ) λ [ A$_{2}$F$_{1}$ ( a, b ; 1 2 ; − sinh 2 µq ) + B (sinh µq ) $_{2}$F$_{1}$ a + 1 2 , b + 1 2 ; 3 2 ; − sinh 2 µq )] (15) a = 1 2 ( λ − √ |E | µ ) ; b = 1 2 ( λ + √ |E | µ ) , (16) E$_{n}$ = − µ $^{2}$( λ − 1 − n ) $^{2}$; n ∈ N , 0 ≤ n < λ − 1 . (17) ψ$_{n}$ ( x ) = ( 1 1 − ( x a ) 2 )$^{λ}$ +1 2 [ A$_{2}$F$_{1}$ ( a, b ; 1 2 ; ( x a ) 2 ( x a ) 2 − 1 ) + B   √ ( x a ) 2 1 − ( x a ) 2   $_{2}$F$_{1}$ ( a + 1 2 , b + 1 2 ; 3 2 ; ( x a ) 2 ( x a ) 2 − 1 )   , (18) E$_{n}$ = µ 2 ( 8 (3 g$_{1}$ − 2 g$_{2}$ ) − ( λ − 1 − n ) 2 ) , n ∈ N , 0 ≤ n < λ − 1 , (19) a = 1 2 (1 + n ) ; b = λ − 1 2 (1 + n ) . (20)