Coefficient-wise Log-concavity of f(μ, x) = Sum[f[k] Pochhammer[μ, 2 k]/(Pochhammer[μ, k+1] k!), {k, 0, Infinity}] at α = 1, β = 2

u = Pochhammer[μ +1, 2 k] Pochhammer[μ +2, 2 (m-k)]/(Pochhammer[μ+2, k] Pochhammer[μ +3, m-k]) (* the term u_k in the summands of the generalized Turanian Δ_{μ}(1, 2; μ) *)

v = Pochhammer[μ +2, 2 k] Pochhammer[μ +1, 2 (m-k)]/(Pochhammer[μ+3, k] Pochhammer[μ +2, m-k])(* the term v_k in the summands of the generalized Turanian Δ_{μ}(1, 2; μ) *)

r = Pochhammer[μ, 2 k] Pochhammer[μ +3, 2 (m-k)]/(Pochhammer[μ +1, k] Pochhammer[μ+4, m-k]) (* the term r_k in the summands of the generalized Turanian Δ_{μ}(1, 2; μ) *)

s = Pochhammer[μ, 2 (m-k)] Pochhammer[μ+3, 2 k]/(Pochhammer[μ+1, m-k] Pochhammer[μ+4, k]) (* the term s_k in the summands of the generalized Turanian Δ_{μ}(1, 2; μ) *)

u1 = u /. {k -> k -1} (* make a shift to u_k *)

I1 = FullSimplify[u1/u]

II1 = I1 /. {m -> 2 k + t}

III1 = II1 /. {k -> q + 2}

v1 = v /. {k -> k -1}(* make a shift to v_k *)

I2 = FullSimplify[v1/v]

II2 = I2 /. {m -> 2 k + t}

III2 = II2 /. {k -> q + 2}

r1 = r /. {k -> k -1} (* make a shift to r_k *)

I3 = FullSimplify[r1/r]

II3 = I3 /. {m -> 2 k + t}

III3 = II3 /. {k -> q + 2}

s1 = s /. {k -> k -1} (* make a shift to s_k *)

I4 = FullSimplify[s1/s]

II4 = I4 /. {m -> 2 k + t}

III4 = II4 /. {k -> q + 2}

g1 = (μ +1) (μ + 2 (q +2)) (μ +2) (μ + 2 (q+t+2)) (μ + 2 (q+t+2) +1)/((μ+3) (μ +q+2+1) (μ +3) (μ +q+t+2+1) (μ+q+t+2+2))

g2 = (μ+2) (μ +2 (q+2)) (μ + 2(q+2) +1) (μ+1) (μ + 2 (q+t+2))/((μ+3) (μ +q+2+1) (μ +q+2+2) (μ +3) (μ +q+t+2+1))

g3 = μ (μ + 2 (q+t+2)) (μ + 2 (q+t+2) +1) (μ + 2 (q+t+2) +2)/((μ +3) (μ +q+t+2+1) (μ +q+t+2+2) (μ +q+t+2+3))

g4 = (μ +2 (q+2)) (μ + 2 (q+2)+1) (μ + 2 (q+2)+2) μ/((μ +q+2+1) (μ +q+2+2) (μ +q+2+3) (μ+3))

f1=Together[g1 + g2-g3-g4] // Factor

n1 = Numerator[f1] (* the term \tilde{A}_k ≤ 0 is equivalent to n1 ≤ 0 with q ≥0, t≥0, mu ≥ 0 *)

f2 = Together[III1 g1 + III2 g2 - III3 g3 - III4 g4] // Factor

n2 = Last[FactorList[Numerator[f2]]][[1]] (* the term \tilde{A}_{k-1} ≤ 0 is equivalent to n2 ≤ 0 with q ≥0, t≥0, mu ≥ 0 *)

{quo, rem} = PolynomialReduce[n2, {n1}, {q, t, μ}]

n2 === n1 + rem (* Note that the coefficients of rem are not all negative *)

CylindricalDecomposition[Implies[μ ≥ 0 &&q≥ 0 &&t ≥0 && n1≤ 0 , n2 < 0], {μ, q, t}] (* Using Cylingdrical Algebraic Decomposition, we deduce that n1 ≤ 0 implies n2 ≤ 0 with q ≥0, t≥0, mu ≥ 0 *)

Coefficient-wise Log-concavity of f(μ, x) = Sum[f[k] Pochhammer[μ, 2 k]/(Pochhammer[μ, k+1] k!), {k, 0, Infinity}] at α = 1, β = 3

u = Pochhammer[μ +1, 2 k] Pochhammer[μ +3, 2 (m-k)]/(Pochhammer[μ+2, k] Pochhammer[μ +4, m-k]) (* the term u_k in the summands of the generalized Turanian Δ_{μ}(1, 3; μ) *)

v = Pochhammer[μ +3, 2 k] Pochhammer[μ +1, 2 (m-k)]/(Pochhammer[μ+4, k] Pochhammer[μ +2, m-k]) (* the term v_k in the summands of the generalized Turanian Δ_{μ}(1, 3; μ) *)

r = Pochhammer[μ, 2 k] Pochhammer[μ +4, 2 (m-k)]/(Pochhammer[μ +1, k] Pochhammer[μ+5, m-k]) (* the term r_k in the summands of the generalized Turanian Δ_{μ}(1, 3; μ) *)

s = Pochhammer[μ, 2 (m-k)] Pochhammer[μ+4, 2 k]/(Pochhammer[μ+1, m-k] Pochhammer[μ+5, k])  (* the term s_k in the summands of the generalized Turanian Δ_{μ}(1, 3; μ) *)

u1 = u /. {k -> k -1}(* make a shift to u_k *)

I1 = FullSimplify[u1/u]

II1 = I1 /. {m -> 2 k + t}

III1 = II1 /. {k -> q + 2}

v1 = v /. {k -> k -1}(* make a shift to v_k *)

I2 = FullSimplify[v1/v]

II2 = I2 /. {m -> 2 k + t}

III2 = II2 /. {k -> q + 2}

r1 = r /. {k -> k -1} (* make a shift to r_k *)

I3 = FullSimplify[r1/r]

II3 = I3 /. {m -> 2 k + t}

III3 = II3 /. {k -> q + 2}

s1 = s /. {k -> k -1} (* make a shift to s_k *)

I4 = FullSimplify[s1/s]

II4 = I4 /. {m -> 2 k + t}

III4 = II4 /. {k -> q + 2}

g1 = (μ+1) (μ + 2 (q+2)) (μ +3) (μ + 2 (q+t+2)) (μ + 2 (q+t+2)+1) (μ + 2 (q+t+2)+2)/((μ +4) (μ +q+2+1) (μ +4) (μ +q+t+2+1) (μ+q+t+2+2) (μ +q+t+2+3))

g2 = (μ+1) (μ+2 (q+t+2)) (μ +3) (μ + 2 (q+2)) (μ + 2 (q+2)+1) (μ + 2 (q+2)+2)/((μ+4) (μ +q+t+2+1) (μ+4) (μ +q+2+1) (μ+q+2+2) (μ +q+2+3))

g3 = μ (μ + 2 (q+t+2)) (μ + 2 (q+t+2) + 1) (μ+ 2 (q+t+2)+2) (μ+ 2 (q+t+2)+3)/((μ+4) (μ+q+t+2+1) (μ+q+t+2+2) (μ+q+t+2+3) (μ+q+t+2+4))

g4 = μ (μ +2 (q+2)) (μ +2 (q+2)+1) (μ + 2 (q+2)+2) (μ + 2 (q+2)+3)/((μ+4) (μ+q+2+1) (μ+q+2+2) (μ +q+2+3) (μ+q+2+4))

f1=Together[g1 + g2-g3-g4] // Factor

n1 = Numerator[f1] (* the term \tilde{A}_k ≤ 0 is equivalent to n1 ≤ 0 with q ≥0, t≥0, mu ≥ 0 *)

f2 = Together[III1 g1 + III2 g2 - III3 g3 - III4 g4] // Factor

n2 = Last[FactorList[Numerator[f2]]][[1]] (* the term \tilde{A}_{k-1} ≤ 0 is equivalent to n2 ≤ 0 with q ≥0, t≥0, mu ≥ 0 *)

{quo, rem} = PolynomialReduce[n2, {n1}, {q, t, μ}]

n2 === n1 + rem (* Note that the coefficients of rem are not all negative *)

CylindricalDecomposition[Implies[μ ≥ 0 &&q≥ 0 &&t ≥0 && n1≤ 0 , n2 < 0], {μ, q, t}] (* Using Cylingdrical Algebraic Decomposition, we deduce that n1 ≤ 0 implies n2 ≤ 0 with q ≥0, t≥0, mu ≥ 0 *)